Abstract
We consider a storage allocation model with a finite number of storage spaces. There are primary spaces and secondary spaces. All of them are numbered and ranked. Customers arrive according to a Poisson process and occupy a space for an exponentially distributed time period, and a new arrival takes the lowest ranked available space. We let and denote the numbers of occupied primary and secondary spaces and study the joint distribution in the steady state. The joint process behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rate becomes large, and the storage capacities and are scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.
1. Introduction
We consider the following storage allocation model. There are primary and secondary storage spaces. The primary spaces are numbered and the secondary ones are numbered . Customers arrive according to a Poisson process of rate , and each customer occupies a storage space for an exponentially distributed amount of time, with the mean occupation time . A new arrival takes the lowest ranked available space. If all spaces are filled, then a new arrival is turned away and lost. The policy of taking the lowest ranked space is called “first-fit allocation.”
We can consider the storage spaces as parking spaces of a restaurant. The primary spaces are in a lot right next to the restaurant, and the secondary spaces are located somewhere further away from the restaurant. Lower ranked spaces will be closer to the restaurant so it is natural for a customer to use the first-fit policy. Since spaces are occupied and emptied at random times, this model is called a dynamic storage allocation model. Design and analysis of algorithms for dynamic storage allocation are a fundamental part of computer science [1]. In such applications we can consider the customers as records, files, or lists and the storage device as a memory device. As time evolves, items are inserted and deleted, and the storage device, which is a linear array of “cells,” will have regions of occupied cells alternating with interior holes. This is referred to as memory fragmentation in computers, and collapsing the holes corresponds to running a defragmentation program.
In the language of queueing theory, the model with finite secondary storage spaces can be called the queue (the Erlang loss model) with ranked servers. The main contribution here is to study the effects of the finite storage capacity, for systems with a large number of both primary and secondary storage spaces and a commensurably large traffic intensity, which we denote by . Thus we study the model asymptotically for with .
We let and be the numbers of occupied primary and secondary spaces, and we will focus on the joint distribution of and , in the steady state. The distributions of both and are readily computed, as these processes behave as Erlang loss models, with and servers, respectively. Thus their steady state distributions are truncated Poisson distributions. However, the distribution of the number of occupied secondary spaces is much more complicated, as is the joint distribution .
We focus here on only the steady state distribution but comment that the transient behavior of the standard Erlang loss model can be analyzed by singular perturbation methods of the type employed here (see [2]). Thus we believe that, with significant additional effort, the transient behavior of the joint process (, ) could also be ultimately analyzed.
There has been much past work on the model with an infinite (secondary) storage capacity () since Kosten [3]. Various aspects of the solution were also studied in [4–7], but the solutions are in a complicated form, which is difficult to evaluate asymptotically for , due to the presence of an alternating sum. We derived the joint steady state distribution of the process in [8] using a discrete version of the classic method of separation of variables. We obtained the solution as a contour integral that involves certain polynomials related to hypergeometric functions. Such representations enabled us to obtain a complete set of asymptotic results including the joint distribution , for [9–12].
The solution of the finite capacity model with seems more complicated than the solution of the model with . But we will show here that a singular perturbation analysis is again fruitful, and we will obtain a complete set of asymptotic results for , which depends also parametrically on , and the numbers and of primary and secondary storage spaces. Most of the time we will scale all of , , , and to be of the same order as the traffic intensity . We will focus on understanding the effects of the finiteness of the secondary storage capacity .
The remainder of the paper is organized as follows. In Section 2 we state the basic equations and briefly describe their forthcoming analysis. In Section 3 we summarize all of the main results, and the joint distribution will have different asymptotic expansions in three main regions of the state space, which is the lattice rectangle . Moreover, there are also various boundary, corner, and transition curves where different expansions will be needed. In Section 4 we derive the asymptotics of the joint distribution in the three main regions, while in Sections 5–7 we treat the boundary, corner, and transition ranges. In Section 8 we will do some numerical comparisons to test the accuracy and robustness of our asymptotic results. Some discussion of our results also appears in Section 8. Since the analysis is quite technical, we have written this paper so that the derivations in Sections 4–7 can be omitted upon a first (and perhaps even later) reading(s).
2. Statement of the Problem
We consider a system with primary and secondary storage spaces (or servers). The primary spaces are ranked and numbered while the secondary spaces are numbered . Customers arrive according to a Poisson process with rate parameter and a new arrival takes the lowest ranked available space, if possible a primary one. If all spaces are occupied further arrivals are turned away and lost. All of the storage spaces are identical and a customer occupies a space for an exponentially distributed amount of time, with the mean occupation (or service) time being . We then let and be the numbers of occupied primary and secondary spaces, respectively. We also introduce a dimensionless parameterto denote the traffic intensity.
The joint process corresponds to a continuous time random walk in a lattice rectangle. Figure 1 indicates transition rates. The steady state distributionis independent of the initial values and and satisfies the following balance equations:The main balance equation (3) applies in the interior of the lattice rectangle and along the boundary , (4)–(6) correspond to boundary conditions along three of the four boundaries of the rectangle, and (7)–(9) are corner conditions. Also, (6) applies at so the corner condition at is . We also have the normalization condition
The process by itself behaves precisely as the Erlang loss model ( queue with servers). This is well known to have, in the steady state, a truncated Poisson distribution; hence
The total number, , of occupied servers also follows a truncated Poisson distribution. Therefore,
We recently obtained in [13] explicit expressions for the joint distribution , but they are not very insightful due to their complexity. Thus we study the problem asymptotically, for with . This means that there are many arrivals but the numbers of storage spaces, both primary and secondary ones, are commensurately large. Note that if with then the probability distribution would concentrate on a single lattice point, with as . Here , , and , . Thus this limit would not be particularly interesting. There are, however, certain cases where either or is large but , that should lead to interesting results, but we do not analyze them here.
We next introduce the parametersand in the present limit we have , , where we view and as fixed as . We may then view the process on a “coarse” spatial scale, withOn the scale the random walk takes small steps () and the state space may be approximately viewed as the continuous rectangle Setting the main balance equation (3) becomeswhich is a difference equation with small differences, of order . The boundary condition along in (4) may be replaced by the “artificial boundary condition”The above is obtained by requiring that (3) holds also at and comparing this to (4). Introducing simplifies some of the calculations, but this quantity has no physical meaning.
The asymptotic structure of the joint distribution will be very different for four main regions in the parameter space. We call these regions and they are sketched in Figure 2. They are defined by the inequalitiesIt will also prove useful to define as follows the curves that separate these four regions:Note that the union of all the sets in (19)–(25) is the entire open quarter plane in parameter space. We purposefully exclude the coordinate axes and , as they would require entirely different asymptotic analyses. The separating curves in (23)–(25) will also require separate analyses, and we will obtain results that apply not only along the curves but also in small neighborhoods of these curves, which will be defined precisely later. This will produce results that asymptotically match to those in the main regions.
The presence of the different regions can be explained intuitively. If () there are enough primary spaces to service all storage requests and the secondary spaces will generally not be needed. If but () the primary spaces are insufficient but the total number of spaces is adequate. Then we might expect that typically all primary spaces and about () secondary spaces will be occupied. If then typically all primary and secondary spaces will be occupied. Then we might expect to be concentrated near , . The further splitting of into the regions and is difficult to explain intuitively in terms of the basic model, but we will explain this dichotomy via our asymptotic analysis. We also note that the asymptotic behavior of the distribution in (11) undergoes a transition when passes through 1, while (12) undergoes an analogous transition when passes through 1. However, neither (11) nor (12) undergoes a transition along . In the analysis that follows we will also need to, for each region of parameter space, separately analyze several different regions of the state space, which corresponds to the rectangle , on the coarse spatial scale. It will sometimes prove necessary to analyze boundary and corner regions where the discrete nature of the model must be considered.
3. Summary of Results
In the analysis it proves sometimes useful to use the variables whereso that (resp., ) measures the number of unoccupied primary (resp., secondary) spaces. Then we also letso that corresponds to the probability that all of the storage spaces are full. In (27) we did not indicate the dependence of and on the parameters , , and .
We begin by giving asymptotic results for .
Proposition 1. For and fixed one has
Here and throughout the paper, we use the convention that corresponds to the union of the open sets and and also the separating curve (cf. (23)). Similar comments apply for and . We refer to the asymptotic limit in (29) as corresponding to , where we now give the precise scaling, , that applies near the separating curve in (24). The results in (28)–(30) will follow from our asymptotic analysis of the joint distribution, but we note that these also follow easily from (12), by setting and expanding the result for and different ranges of (thus ).
It will prove convenient to express some of our results in terms of the three constants , , and ; these depend only on the parameters , , and . We summarize below the leading order asymptotics of these constants.
Proposition 2. Define the constants , , and by the relationswhere is the Airy function and its maximal root
We note that the relation holds for all cases of the parameters. In (38) we have thus defined the precise scaling near the separating curve , as . Note that is not defined for region while is not defined for , as then the corresponding constant will play no role in the analysis. We conclude by giving the precise scaling for (near ), which will beNote that (46) can be predicted from the marginal distribution in (11), as the sum in the denominator undergoes a transition for , which is the same scaling as in (46).
3.1. Joint Distribution and Its Limits
Now we consider the joint distribution; for . We recall that and are the scaled numbers of primary and secondary spaces. The state space of the random walk is the lattice rectangle in Figure 1, and on the coarse spatial scale this can be viewed as the continuous rectangle . Our goal is to give a complete asymptotic description of the joint distribution for , including ranges of the state space where there is appreciable mass and also ranges where is asymptotically small. This corresponds to the tails of the distribution and in such ranges is typically exponentially small for large . We first discuss the ranges where there is significant mass, and this will lead to certain limiting distributions, which will be very different for regions of parameter space in (19)–(25).
Proposition 3. For one has the following limiting distributions:(i) (thus ) which can be recast as the limit(ii) (thus ) and this applies for and (i.e., ).(iii) (thus ) which holds for and .
When we have so the secondary storage spaces will be rarely needed, and then approximately follows the Poisson distribution in (47), which has also the Gaussian limit in (48). The results in (47) and (48) provide no information on for , but later we will estimate precisely these probabilities. We also note that when , (47) ceases to be valid for , for then if almost all primary spaces are full there may well be some secondary spaces also occupied, and thus may become comparable to for , for this range of . If and the primary storage spaces are insufficient to meet the demand, but the total number of spaces does suffice. Then (49) shows that primary spaces and secondary spaces will tend to be occupied, with the joint distribution being a product of a geometric and a Gaussian. This also shows that, to leading order for large , the processes and decouple. When we have and the totality of storage spaces is not enough to meet the demand. Then typically all but a few spaces, both primary and secondary, will tend to be occupied, with the numbers and of available spaces following the discrete joint distribution in (50). From (50) we can easily show that so that the total number of empty spaces is geometrically distributed; this result also follows easily from the exact expression in (12). We will later see that the tail behavior of (50), for and/or , is quite different according as or , which again will indicate that the triangle in parameter space needs to be split into the two regions and .
We next study the transitions between the three limiting results in Proposition 4.
Proposition 4. For one has the limiting distributions: (i), , and (thus and ) where is the parabolic cylinder function of order and argument . When the above simplifies to(ii), , and
As the truncated Gaussian distribution in (55) approaches the free space Gaussian in (49), which applies for . For , (55) asymptotically matches to (50), when the latter is expanded for and simultaneously , with the product held fixed.
The complicated distribution in (53) is a necessary intermediate result since (47) and (49) do not asymptotically match. The right-hand side of (53) is of the form (density in ), with the density having support in the quarter plane , . Thus if there will tend to be empty primary spaces and full secondary spaces, with now an intricate coupling between the processes and . Finally, we note that the results in items (i) and (ii) in Proposition 3 and in item (i) of Proposition 4 are independent of the secondary storage capacity , while item (iii) in Proposition 3 and item (ii) in Proposition 4 do depend upon .
3.2. Joint Distribution: Main Regions of State Space
The asymptotic expansion of will be different for the four parameter ranges indicated in Figure 2 and also for three main regions of the state space, which we call , , and , and we define/discuss these below.
First consider region of parameter space, so that , and define the curveThis curve depends on both and and thus on both of the total numbers, and , of primary and secondary storage spaces. The curve is defined for and we haveFor region (and indeed also for and ) we have so that (56) connects the point to the corner point in the scaled state space. The curve divides the state space into the two regions and , withHere we defined as an open set, while is bounded by the four curves , , , and and we include only the third of these as a part of . This is because the asymptotic expansion that will apply in the interior of will remain valid near , but not near the other three bounding curves. The expansion valid in will break down if either , , or . We sketch in Figure 3 the curve and we recall that if most of the mass in is concentrated in the range and (see Proposition 3), and this corresponds to the lower bounding curve for .
Proposition 5. For () the asymptotic expansions of are as follows: (i) where are related to via the mapping, for , where is the Jacobian associated with (62); that is,(ii) where is given by (45) for region where are related to by where with with corresponding to , and where is the Jacobian associated with (68), so that
We can view (68) as representing a family of curves in the plane, with indexing the family and increasing along a curve. When the curves in (68) meet at the corner point and we also note that the Jacobian in (71) vanishes when , indicating a singularity in the transformation in (68). When the curve becomes the horizontal segment . But then (69) shows that so that in (67). Thus near the expansion in (68) becomes invalid. When the curve in (68) becomesand eliminating we see that (72) is precisely the curve in (56) that separates from . For the curves in (68) fill a portion of , but then the leading term for is given by (60), and (65) corresponds to only an exponentially small correction to (60). When the curve in (68) is tangent to the line at the point , which will have significance for the parameter region . We also note that for when the curves in (68) hit the -axis (then by (68)) and then the first factor in (70) becomes singular, which indicates that the asymptotics become invalid. Along corresponding to , in (70) is again singular. Thus (70) is singular when , , and , corresponding to the three curves (, , and ) that bound the region . We will give the appropriate expansions near these bounding curves in Section 3.3.
For the curves in (62) fill the entire region , with corresponding to the line segment , , and corresponding to the curve (then (62) coincides with (72)). When we have and the curves in (62) hit the line at finite and nonzero slopes, for all . As increases each curve will hit first either the -axis or the -axis. When the -axis is hit first. For this occurs at a finite value of , when , but if in order to approach the -axis we must let and in such a way that is held fixed. In this limit (62) may be approximated by and . We discuss in more detail the behavior of (60) as later, when we give the asymptotic expansion(s) for that apply for and . When the curves in (62) hit the -axis when , for then . In particular if the corresponding curve hits the -axis at in (57). Near both the - and -axes, (60) will have singular behaviors and other expansions must be constructed. Note, however, that (60) is not singular along the curve , whereas (65) is singular. We can simplify (60) near , and then we obtain the more explicit form which holds for and . However, we note that since we have and thus in (73) is exponentially small in . This is true for the entire domains and , as there is very little probability mass in these ranges if .
We next consider regions and in parameter space, where it will become necessary to break up the state space into the three regions , , and . These regions are sketched in Figures 4 and 5. The curve that separates from is again given by (56), while the curve separating from will beand thus, since now ,
We thus define asand, for regions and , now becomeswith still defined by (59).
Proposition 6. For the asymptotic expansions of are as follows: (i) where with and is the maximal root of the Airy function . In (79), for , is given by (35) for (then ), and is given by (36) for .(ii). The expression in (60) applies with for , is given by (32) for , and is given by (33) for .(iii). The expression in (65) applies with given by (45) for , is given by (44) for , and for .
In contrast to regions and , the expansion (79) in is a completely explicit function of , and . We also note that has a simple linear dependence upon , and . In Section 4 we give a more geometric interpretation of this expansion, and we also observe that the form of (79) is slightly different from the expansions in and , as the former contains an additional factor that is of order , and thus gives an additional subexponential dependence on . While the forms of , , and change according to whether lies in the regions , or of parameter space, the ratios : : remain the same for these three cases.
The expansion in (79) is valid only in the interior of . As there is a singularity due to the factor in in (83). For there is also a singularity due to the factor in (83), and as , we find that . The curve that separates from corresponds to , and along the factor vanishes. Thus (83) shows that which also indicates a nonuniformity in the asymptotics. Later we will give appropriate expansions near the three bounding curves (, , and ) of region .
In Figure 4 we also indicate the curvewhich lies entirely within (when ) and corresponds to , where is in (61). We can also show that so that, for a fixed , achieves a local maximum along . Note that corresponds to ) in (62), for then and .
For the curve in (84) plays no role, as it lies outside of , but now the curvelies within , connecting the points and . Along we have and then and , so that (68) becomes and . Then for region , will have a local maximum along in .
For most of the mass will lie near the corner point , where and meet, but neither (60) nor (65) (with the appropriate and ) are valid there. For , will be maximal near the point , and (73) applies for (or ) for both regions and . By expanding (73) about , which is where is maximal, we obtain precisely the expression in (49).
Next we consider region of parameter space. Now the state space will be split into and , and will be absent. This is sketched in Figure 6, and we also observe that as in region , the curves and in (75) and (56) become identical, and thus shrinks to this curve. For there is a new curve that comes into play; namely,We have and the curve hits the -axis when The curve now separates from and corresponds to in (68), and also in (81).
Proposition 7. For (thus ) the asymptotic expansions of are as follows: (i), , . The expansion in (79) applies with now given by (41).(ii). The expansion in (65) applies for with .
Again, different expansions must be given near and near all boundaries of the state space. The curve in (85) still lies within and is sketched in Figure 6, with again along this curve.
We conclude by noting that, for all regions of parameter space , the expansion in depends upon the secondary storage capacity (or ). For regions and the expansions in and ( only) are independent of , except through the curve that bounds . Thus if the effects of the finite storage capacity appear in only, and then letting will recover the results for the storage model in [10, 11], which assumes that . For , hence , the expansions in and depend upon only through the multiplicative constants and , which now depend on in view of (33) and (36). For , hence , again the result depends on only through the constant (cf. (41)). For , that is, along and near the curve , we have so all four of these curves coalesce. A special analysis is required for for values of near this curve(s).
This completes our summary of the asymptotic expansions in the three main regions, , , and , of the state space.
3.3. Joint Distribution: Boundary, Corner, and Transition Regions
We next analyze the four boundary segments of the state space rectangle, namely, the line segments , , , and . As we previously discussed, the expansions in the three main regions become invalid near the boundary of the state space, with the exception of the expansion in (60), which remains valid near , reducing to (73) when , at least for . For the four boundary segments we will typically consider the respective scales , , , and , though at times a different scaling must be considered in addition to these discrete scales. Note also that a particular point on a boundary segment may also require a separate expansion, and this occurs in , when hits at the point . Also, in , the curve that separates from hits the -axis when and this point requires a separate analysis. Finally, in the curve hits the -axis when .
After treating the four boundary segments we will give asymptotic results that are valid near the four corner points, , , , and , of the state space. Finally we give results for points that lie on or near the transition curves (for ), (for ), and (for ). Unlike the previous subsections where we listed the results by region of parameter space, here we go by region of state space, and each proposition will correspond to one such region and the different results for the different will be collected in that proposition.
Proposition 8. For and , , one has the following expansions: (i), (ii), , (iii), (iv), , where is the imaginary axis in the -plane if , and if the contour is to be indented to the right of the pole at . Here is the gamma function. When the expression in (94) simplifies to(v), , where , so that corresponds to in (81), is defined in (82), and can be computed from (80) by setting and replacing by . The value of in (96) is given by (34) for , by (35) for , and by (36) for .(vi), . The expression in (96) applies with now given by (41). For region (hence ), (96) holds for , where can be computed from (37).(vii), where is the solution of the algebraic equation that satisfies and , so that and , and For and one can obtain more explicitly, with(viii), where
We note that the regions of validity of the expansions in Proposition 8 are such that the four corner points of the state space are excluded. For regions and the expansions for are independent of the secondary storage capacity . For region the result in (96) does depend upon , but only through the multiplicative constant , which now depends on in view of (36). This is also true for region (and ) when , as then is given by (41) or (37). However, for and , the expressions in (98) and (105) show that now depends in an intricate way on . Thus the finiteness of the secondary storage capacity affects the probabilities that there are even a few secondary spaces occupied.
For region , is exponentially small for and approximately follows a Poisson distribution, as indicated previously in Proposition 3. The results in (88)–(93) better quantify (47) and estimate the exponentially small error from the Poisson approximation. For regions , , and , is always exponentially small for , and the approximations do not distinguish from . By evaluating the contour integral in (105) for we can verify that (105) asymptotically matches to (96) (with given by (41)), as . Similarly, for (105) will match to (98) as .
Next we consider points near , with . Note that only regions and can be bounded by , for .
Proposition 9. For , one uses the scales and , so that denotes the number of empty primary storage spaces. The expansions are now the following: (i), ; , where is given by (34), (35), and (36) for parameter regions , , and , respectively, is obtained by replacing by in (82), and(ii) with , where and is again given in Proposition 2 for the different ranges .(iii) or : or . Now the expressions in (109) and (111) apply over all , with given by (37) for and (41) for .(iv): , where , so that , and is given again in Proposition 2 for the three cases , , and .(v): or , where is given in (74) and has the expansions in (31)–(33).
Note that the asymptotics are most complicated in (112), which occurs when and this is where the curve hits the line . This intersection occurs only for regions and . The integrand in (112) is a meromorphic function of , having double poles at all roots of the Airy function .
Proposition 10. For and , , one has the following expansions: (i) where is the root of the cubic equation that satisfies and The expansion of is given by (31)–(33) according to subregions of .(ii): or where and the expansion of is given in Proposition 1 for the three cases , , and (when ).(iii): where
We see that for each of the three cases the dependence of on is of the form (times a function of or ), where the geometric ratio depends upon and undergoes a transition when increases past , which can occur only for regions , , and . We note that and , so that is continuous along , with , as indicated in (121). The expansion in (117) develops a singularity as , in view of the factor . The expansion is also singular as (for all regions ) and as (for region only). Then we are approaching the corner points or of the state space. Using the expansionswe can easily show that (121) matches to (117) (for region in the intermediate limit where and ) and to (114) (with now and ). Note also that the ratio is asymptotically the same for each of the three regions , , and , in view of (28)–(33). For any region , for and , is exponentially small in .
Proposition 11. For and one has (for all regions in parameter space)where is given bywhere is given in (69), has the respective expansions in (28)–(30), and
Thus (127) gives the expansion when there are only a few secondary spaces empty and a fraction of primary spaces occupied. The expressions can be simplified in the limits (then ) and (then ), but then other “corner” expansions will apply. Note that (127) is a completely explicit expression in terms of and .
Next we examine the four corners of the state space, where , , , and . We recall that these ranges are important in that for region most of the mass concentrates near the corner , while for region most of the mass occurs near . We will start with the corner and proceed about the perimeter of the state space in a counterclockwise manner.
Proposition 12. For , , and any parameter region one has where is given in Proposition 1, is in (69), and
This gives the expansion when there are but a few primary spaces occupied and a few secondary spaces empty. Next we consider the corner point , and this will typically correspond to . The results will be very different for region compared to those for the remaining regions.
Proposition 13. For and one has (i)(ii), where is as in (94), and when (iii) where is given by (34), (35), or (36) for , , or , respectively.(iv)(v); , ; ,
We note that (136) and (137) are continuous along the curve , corresponding to , in view of the expansion in (36) for in . Thus the leading term for for can be obtained by either using (36) to compute in (136) or replacing by in (137). The scale must be considered (cf. (138)), as the approximation in (136) cannot directly match to those for , (cf. (114)) or , (cf. (96)). We can view (136) as a special case of (138), letting in the latter and noting that For region we see from Figure 6 that only state space region meets the corner . Then (137) matches directly to (98), in the limits , , and to (117), in the limits , . Note that, for region , , which follows from (118) (this is true also for , but then (117) applies only for and thus not near the corner ). We can obtain a result analogous to (138) for the transitional range , but we do not give that for the sake of brevity.
Proposition 14. For and , or for and , the expansions of are as follows: (i), (ii), where is the minimal root of the parabolic cylinder function ; that is, , and .(iii), , and , with and which when simplifies to(iv), , , (v) with now and , , , (vi), and is given in Proposition 2 for the different regions .
For region , (141) show that the dependence of on and is quite simple, but we have to distinguish the cases and . The same is true for regions , , and , where (149) applies for all , except now the form of in (149) is different for the five cases in Proposition 2. However, when , the asymptotic structure of is quite complicated, and we must consider separately the scales and . For we obtain the limiting density in (53) or (143), as the limit of for . This expansion applies for , for any , but becomes invalid as . For the asymptotic behavior of the contour integral in (143) is determined by the singularity at , and the density behaves as for and . This corresponds to either an integrable singularity or a zero of the density (unless then ) and in either case indicates a nonuniformity in the asymptotics. Thus we need the expansion in (145) for . For we have, by Stirling’s formula, and then (145) matches to (143) in the intermediate limit where but . The expansion in (145) itself breaks down when , since by the definition of we have as . Then for we have the expansion in (142), which holds for and we note that so that .
We can show also that the expansions for (cf. (141), (142), and (149)) match in appropriate intermediate limits. Consider (142) for (then we are moving into the range ). We now have and more preciselyso the approximation to the minimal root of the parabolic cylinder function involves the maximal root of the Airy function . Thus, for we havewhich can be obtained by approximated by Airy functions in the double limit where and , with . Using (154) in (142) and expanding by Stirling’s formula (since now ) for a fixed , (142) becomesThen (155) must agree with the expansion of (149) as . Then we use the fact that in region . Also, as , ,Using (156) we see that as , (149) becomesBut by using (153) in (155) we obtain again the expression in (157), which verifies the matching.
Now consider (141) for and (142) for . We now haveso that is exponentially small and is exponentially large. The last factor in (142), namely, , can be approximated by if and is equal to if . Now also and (142) becomes, for , and the above clearly agrees with (141), when we expand these for .
The results in (146)–(148) assume the scaling , and these are needed to asymptotically connect the parameter ranges and . It is only near the corner that we must consider this scaling (and indeed also the -scale). For other ranges of and , we can get the expansions of as limiting cases of other expansions, as they lie in the asymptotic matching range where but . For a fixed and we havefor with we haveand for the scale we have so the last factor in (161) may be approximated by . We can easily verify that as , (159) matches to (146) as . Note that in this limit we have For we have and then (146) clearly matches to (160) and in fact contains the latter as a limiting case. When we have, for ,for , , we haveand for We can again easily verify that (147) matches to (164) for and and to (165) for and . Recalling that , we have and this agrees with (166) for , which verifies the matching between (165) and (166). Thus we have given for all ranges of for and the scaling , . However, only for the range do we get the new results in (146) and (147).
Next we examine the corner , so both primary and secondary spaces will be nearly full. For regions most of the mass is in the range. We will need to consider the scales and also . Consider .
Proposition 15. For and one uses the variables and , and the expansions are as follows: (i), , where are defined in (50) and the expansions for are in Proposition 1; in particular for region one has and then (168) is a proper discrete distribution over the range(s) , .(ii), , , where and are the roots of , ordered as .(iii), , (iv), , , where and is given by (171) in terms of .
When most mass concentrates on the scale so there tend to be but a few available primary and secondary spaces. For , the result in (168) still applies but now is exponentially small (cf. (30)). For we have . Later we will study the behavior of the contour integral in (168) as and/or , and we will see that for we can get exponential growth in certain sectors, such as if with . From Proposition 15 we also see that the probabilities of finding empty primary spaces and secondary spaces are quite complicated, and their estimation involves contour integrals of Airy functions. These probabilities are however quite small, in view of the factors and in (169) and (173). In (173) we can use (175) and rewrite the result in terms of and , thus eliminating .
Finally we give results that apply near the transition curves that separate , , and . Note meets for region(s) , meets for region , and meets for region (see also Figures 3–6).
Proposition 16. For , , and one haswhere is given by (56), and has the expansions in (31)–(33) for the different regions .
For we are moving into region and the integral in (176) approaches , and then (176) becomes simply the expansion of the result in (60), about . Thus the matching of (176) to is immediate, and we can also verify the matching to (65), by expanding the latter as , and (176) for . We recall that Proposition 2 shows that the expansion of is the same for regions , , and . The function in (178) is obtained by setting and then in (62). The function is obtained by setting and in (64) and corresponds to the values of this Jacobian along the curve . Note also that can be obtained by setting and in (68), and corresponds to the Jacobian in (71) along the curve , with .
Proposition 17. For , , and one haswhereand has the expansions in (31)–(33) (one can replace by in (188)), and is given in (75).
For we can expand the integral in (188) by the saddle point method, after shifting the integration contour far to the right, with . Then the Airy function in the integrand may be approximated usingand we can verify that (188) for matches to the result in (60), as . For the behavior of the integral in (188) is determined by the singularity with the largest real part, which is the double pole at . Then standard singularity analysis can be used to show that (188) for agrees with the expansion of (79) as . Note that in this limit becomes and proportional to .
Proposition 18. For , , and (with defined in item (vii) of Proposition 8) one haswhere is defined in (86).
Thus the transition from to involves a slightly different integral (cf. (191)) compared to the transition from to (cf. (188)), as the former has simple poles at the Airy roots. Using standard asymptotic analysis we can show thatand (192) can be used to verify the matching between (191) and the result in (65), where in . Similarly (193) can be used to verify matching to the result in (79), with now given by (41). The factor involving in (191) can be expanded in Taylor series, similarly to (189), by replacing by and by .
As we approach (where ) from within we see that (191) develops a singularity, in view of the factor . As we approach from within , the expression for in (183) vanishes and that for in (184) develops a singularity. Thus the expansions in both Propositions 16 and 18 become invalid, and below we give a new result that applies for , where meets .
Proposition 19. For , with and ,where and is in (86) and in (75).
With the scaling , the curves , , and nearly coincide, with the differences being . In (194) we can again replace by its Taylor expansion and replace everywhere by , thus writing the result in terms of and (or ), along with and . Note that is still defined by , which differs from by an amount that is . Thus, for region the transition range in state space, between and , involves a somewhat more complicated integral than those in Propositions 18 and 19.
This completes our summary and discussion of the various regions of state space, which carry zero volume in space but are necessary since the results in , , and do not always apply.
4. Asymptotic Expansion in Region
In this section we will construct the expansion for , that is, (65). The analysis for the complementary regions and is virtually identical to that for the model with an infinite secondary capacity , and the detailed calculations can be found in [10]. Here we will only discuss, for regions and , those aspects that change when . In order to uniquely determine the expansion in , we will need to use asymptotic matching to the corner expansion that applies on the scale , , and this is discussed in Section 4.2. For parameter region (and also for ) we will also need to carefully analyze the scale and , which will be necessary to determine the multiplicative constant that arises in the asymptotic expansion in ; this analysis is done in Section 4.3.
4.1. Ray Expansion in the Interior of
We analyze the scaled equation in (17) using the ray method of geometrical optics, where we assume an expansion of the formThen we haveand using (195) in (17) we obtain in the limit as the “eikonal” equationand at the next order in the “transport” equationThe first-order PDE in (197) can be solved by the method of characteristics (see [14]), where one must solve the five ODEsHere is a parameter along a given characteristic curve, which is also called a “ray,” due to applications in optics.
To uniquely specify the solution to (197), we must either specify along some curve, called the “initial manifold,” in the plane, or use a singular solution, where all the rays emanate from a single point. The appropriate solution to the eikonal equation must be determined for each individual problem, and we will see that for the present model we will need to use three different solutions to (197), with two having the boundary as the initial manifold and the third corresponding to all the rays emanating from the corner point . The first two solutions will correspond to regions and and the third to . Since and arose also in the infinite capacity model, where , we discuss these only briefly, and the details of the corresponding solutions to (197)–(201) can be found in [10].
Let us denote the solution in region as to distinguish it from (195), which will apply in . If this expansion will satisfy the boundary equation along (or ) in (4), or, equivalently, (18), we must haveRequiring to satisfy (202) is equivalent, up to an additive constant in which can be incorporated into , to specifying along the initial manifold . Solving (199)–(201) subject to (202) leads toand then the rays are given in parametric form by (62), which relates to . When we have so that is the point where a ray hits the line . We use now instead of as the parameter along a given ray, and is used to index the family. Finally, solving (200) with replaced by leads to the expression in (61). We also note that when we havewhich is an explicit function of . From the first expression in (62) we find that for It follows that for the rays that start from enter the state space, where , for only if , or . If this condition holds for all and hence the rays fill the region indicated in Figure 3. If and then the condition holds only for in the interval . Then these rays fill the domain indicated in Figures 4 and 5. But if and , which is true for parameter region , the condition never holds and then this ray expansion plays no role in the analysis (see also Figure 6). Once we compute in , we can integrate (198) to obtain in (63). Thus we have shown that (63), up to the constant which we have yet to determine, holds in the portion of the state space, for parameter regions .
Now we observe that if , is maximal at and by evaluating also at and we find that near the corner with and , and we haveThe scale for must be analyzed separately and the details are carried out in [10]. For and the result in (141) applies, and by asymptotically matching this to (206) (for ) we conclude thatIf and then in (204) is maximal at , which lies in the range precisely for parameter region . Expanding in (61) about and leads toFor parameter region the expansions in and will be uniformly exponentially small in , as there is little mass in these state space ranges. Thus the main contribution to the double sum in the normalization condition (10) will come from and in particular from the scale and (corresponding to ). Then normalizing the approximation in (208), after approximating the sum over by an integral over , we conclude that , and hence (207) applies also for parameter region (and, by continuity, the relation holds in () also).
We have thus shown that the ray expansion in has the state space regions and as “shadows,” and thus the other solutions to (197) must apply. To construct the solution in we must slightly modify the ansatz in (195) and now expand the joint distribution asNow will satisfy (197) and will satisfy (198), and the subexponential term will satisfy the PDEAgain the detailed analysis can be found in [10]. We now find thatand the rays are given in parametric form bywhere we recall that . When we have and , but unlike the rays in , those in are all tangent to the boundary . Thus the boundary is a “caustic boundary,” for for region and for all with for region . The solution to (197) is now given byFor region we can invert transformation (212) and write and explicitly in terms of and . This leads to the expression in (81) for , and then (80) gives in terms of and .
Equation (210) implies that is constant along a caustic ray, so we write . To determine and also completely determine in (209), we need to construct two “nested” boundary layer corrections to the expansion in (209), near , corresponding to the scales () and (). The boundary condition in (18) can only be imposed on the expansion that applies for . Using asymptotic matching between the expansions for , , and the ray expansion in (209) allows us to determine (209) up to the multiplicative constant .
To determine we need to, for regions and , first relate to the constant in the expansion. This can be done by analyzing the transition curve , with the scaling . The details are again presented in [10], and this leads to the conclusion thatIn we have so that , but we have yet to determine either or for region . The conclusion in (214) can also be reached by constructing an expansion near the point , with the scaling and (see subsection in [10] for that analysis). Note also that the curve corresponds to the ray with in (212) and also the ray with in (62). Then we have so that the exponential factors in the expansions in (60) and (209) are continuous along , which separates from .
The region is a shadow of both the and rays for parameter regions and a shadow of the rays for parameter region . For , the caustic rays that fill correspond to in (212), and corresponds to the curve in (86), which separates from . To fill the shadow and thus obtain an approximation to for , we must use a singular solution to (197), one that has all rays start from the corner point . To construct this solution we first integrate the two ODEs in (201), which yieldswhere and are constant along a given ray. Evaluating the eikonal equation in (197) along the corner and using (215) with givesand this leads to the relation between and in (67). Here we take to correspond to when a ray starts from the corner point . Using (215) we solve the two ODEs in (199), subject to and , and we thus obtain the expressions in (68), which give the corner rays in parametric form. Using (215) and (68) we then integrate (200), and choosing for convenience, we obtain (66). Note that can only be determined from (197) up to an additive constant, but such a constant can be incorporated into in (195).
We can view as indexing this family of rays. If is such that then in (216) and (67) becomes infinite, and this corresponds to in (69). This critical ray corresponds to , and for we have , so the ray is the upper boundary of the state space rectangle, where all secondary storage spaces are full. The ray (for corresponds to the curve in (86). For regions we have and then the ray with is the curve in (56). Thus for regions it suffices to consider in the range to fill region in state space. We also note that in order for the rays, which all start from , to enter the state space we need and , which implies that and . Adding the last two inequalities implies that and this is certainly true if .
We next solve the transport equation (198) for . This equation can be written as an ODE along a ray, withwhere we used (199), and the last equality in (217) follows by differentiating (199) with respect to . Introducing the Jacobian associated with the mapping from to variables, after some calculation we find that (217) becomesand the most general solution to (218) is given byHere is an arbitrary function of the parameter that indexes the rays and is thus constant along any particular ray. We have thus determined the expansion in (195) up to the constant and the function . To complete the ray expansion we will need to use asymptotic matching to a local expansion valid near the corner . This is constructed in Section 4.2. In order to accomplish the matching we will need the behavior of (195) as , , and this is examined next.
We note that corresponds to and in this limit the Jacobian in (71) vanishes, andFor we also haveso thatHere we recall that and , and (222) gives the slope at which the ray indexed by hits the corner point .
We can thus invert the transformation in (68) locally, with (222) corresponding to a quadratic equation for , and henceThen we define by replacing by in (67). Note that the right-hand side of (223) approaches if , approaches as , and is equal to if . For we also have withso that . We have thus shown that as we havewhere was approximated by (223), and by (221) we haveso that (225) becomes an explicit function of and .
4.2. Analysis of the Scale ,
We use the variables and and obtain an approximation to that is valid when all but a few of the primary and secondary spaces are occupied. This is likely for parameter regions and but unlikely for and . We define byWe thus denote by the exact probability that there are (resp., ) empty primary (resp., secondary) spaces, and we will later denote by the leading term in an asymptotic expansion of this probability, which is valid for the scale . Then clearly so in asymptotic relations involving we can drop the subscript .
Writing the balance equations in (3), (4), and (5) in terms of leads toand the corner condition in (9) becomesNote that near this corner only the boundaries and of the state space rectangle are relevant, and the problem in (228)–(231) corresponds to a random walk in a quarter plane, in space. For the problem in (228)–(231) may be further approximated byHere in (232)–(234) is understood to be the leading term in an asymptotic expansion of , for , and the corner equation in (231) must also be satisfied by this leading term.
We introduce the double generating functionand from (231)–(234) we obtainFrom (236), by setting we obtainand if we find that . Setting then in (235) leads toThe expression in (238) shows that the total number of empty spaces follows asymptotically a geometric distribution if , and this could be also deduced from (12), by expanding the exact (truncated Poisson) distribution for . Note, however, that (238) holds also for . The factor in brackets in the left-hand side of (236) vanishes when , where are given in (51). Requiring to be analytic at the smaller root determines asThen using (239) in (236) and partially inverting the transform in (236) lead toand hence by the Cauchy integral formulawhere the contour is over a small loop about .
When (, region ), most primary and secondary spaces will be full, and then (241) becomes a normalized discrete distribution withleading to the limit law in (50).
For regions , (241) still represents a local approximation to , but now different arguments must be used to determine , and indeed now will turn out to be exponentially small for large . We proceed to relate (241) to the ray expansions in , , and , noting that, for parameter region , and both border the corner point, while, for , and border this point. We will need to expand (241) asymptotically, for and/or . The integrand in (241) has branch points at and possible poles at solutions of . Now, for but is not a pole in view of the factor in the numerator. The only other possible solution to occurs at and this is a pole ifBut (243) holds precisely when , which is true for regions . Thus for the pole is absent and along the transition curve (cf. (25)) the pole coalesces with the lower branch point, both being at (). We can recast the integral in (241) by using the conformal map , so that the inverse is and (241) becomeswhere is a small loop about , withand the integrand in (245) has a pole of order at . Below we collect some asymptotic results for that will be used in the matching calculations.
Proposition 20. For and/or the function has the following asymptotic expansions:
(i), (ii) with , , for (iii) with , for where(iv), (with ), for (v), , for region . The expression in (249) holds for all and (247) holds for with .(vi), , for region (vii) with , for region , with ,
The results in (247)–(255) may be obtained by expanding the integrals in (241) or (245) by a combination of singularity analysis and the saddle point method. Good references on asymptotic expansion of integrals are the books [15–19], but since these methods are now well established, we merely sketch the proof of Proposition 20.
To obtain (247) we approximate the integrand in (241) for , scaling and using Note that (247) is not only asymptotically true but also an exact expression when , since satisfies the boundary equation in (234). Turning to (245) we see that the integrand has a simple pole at and saddle point(s) where and (257) is equivalent to the quadratic equation One root of (258) is given by in (250), and the complementary root, with a minus sign in front of the square root, will correspond to a second saddle which will not play any role in the analysis. The pole and saddle , whose location depends on the ratio , coalesce when , for regions , , and . Note also that as we have and that the integrand in (245) has a zero at . For region and we have the ordering and we note that is always true. Then we dilate the small loop about to the saddle point contour , which is a circular contour that traverses the saddle in the steepest descent directions, which are . But in doing the dilation we must take into account the contribution from the residue at the pole , in view of (259). It turns out that the residue dominates the saddle point contribution, and we thus obtain the expression in (248). For we have the ordering and then the pole does not contribute as we dilate to the saddle point contour. Setting we use for (245) the standard saddle point estimate and this leads to the expression in (249). The transitional result in (251) corresponds to and then we have , so the saddle is close to a simple pole. Such situations are discussed in detail in [15], and by expanding the integrand in (245) about we ultimately obtain a simpler integrand that is related to , the parabolic cylinder function of order , which can in turn be expressed in terms of the standard error function, leading to (251). Note that as we approach the region and then (251) reduces to (248).
The pole at and zero at coalesce when , which is precisely the curve which separates from in parameter space. For we have the orderingThen for any () we can deform into the saddle point contour and obtain (249) as the approximation to . The limits , , and , require a separate analysis. For the latter becomes close to and we again obtain (247), while for the former the saddle gets close to zero at . By expanding the integrand near , setting , we obtainwhere Evaluating the integral(s) in (264) we obtain (253). Note that the coefficient of the term in the integral is zero, so the result is .
When the pole at and zero at are close together. If so that remains fixed and positive, the saddle lies well to the right of these, and then (249) holds. But if with then the saddle, pole, and zero are all close. Then we must reexamine the integrand in (245) and expand it about . Again setting and replacing by (with ), (245) becomes asymptotically where is a vertical contour in the complex -plane, which is to the right of the pole at . The integral in (266) may be evaluated as a combination of a Gaussian and an error function, and this ultimately leads to (254) with (255). Since the large parameter appears explicitly in (254) and (266) we can define the asymptotic limit more precisely, as with , , and .
With Proposition 20 we are now ready to relate the corner approximation to the various ray expansions, via asymptotic matching. First consider parameter region , so that and come together at the corner point . In the function can be expanded in Taylor series about and as we obtainwhere we used and . Near , the curve that separates from has the slope so that can be approximated by the straight line , which is the same as . Then meets the corner in the sector , and this corresponds precisely to where the asymptotic result (248) applies. Comparing (267) to (248) we see that the matching is possible, if and are related byand this holds throughout . For we have previously determined that and thus (268) determines , as in (30). For region we have and then (268) leads to (33). For , corresponding to , the approximation in (208) holds for and , but now the upper boundary lies within this range. Then applying the normalization condition in (10) to (208) leads toand then is given by (33). Then can be computed from (268), and since , we obtain the expression in (29). We have thus determined both and for all cases of the parameters .
To match to the ray expansion in we note that, for regions , meets the corner over the sector (), while for region this holds for all (). We recall that for the curve separates from , but this curve has infinite slope at and is thus tangent to the line at . So if we approach the corner along any straight line (excluding slopes and ) we are approaching from within . By the asymptotic matching principle, the expansion of for should agree with the expansion of as , . Thus in we must compare (249) to (225). But from (250) and (223) we see that and from (67) we obtainIt follows that and the matching is possible ifUsing (258) to express in terms of , after some calculation we find thatThen choosing we haveThen we use (273) to simplify (272) toso we have determined the functional form of , and (275) along with (219) leads to (70). We have thus completely determined the ray expansion in , as is known for all parameter regions via (274).
It remains only to determine the constant in the caustic ray expansion, for regions and (for , can be inferred from (214)). For this we must relate to by asymptotic matching, but this matching will require the analysis of another scale, which is intermediate to the and scales. This analysis is carried out in the next subsection.
4.3. Analysis of the Scale ,
We will consider and . Then and so we are examining the vicinity of the corner point along parabolas, where is constant. We note that the scale (with and ) was also important in the analysis of the model with (see [10]). There, a more geometric interpretation is given, in terms of caustic rays and caustic boundaries and also in terms of sample paths of large deviations. Since the caustic rays in region cannot fill the entire domain (for parameter regions and ) we would expect a boundary effect near or . The corner scale that we analyzed in Section 4.2 is insufficient for fully understanding this boundary effect, and hence we analyze the scaling , which will connect the cases and , . We cannot give an a priori probabilistic argument of why this scale is needed but mention that it leads to an interesting PDE, namely, (285), and such problems arise in many other areas, including queues with time-dependent arrival rates (see [20]) and steady two-dimensional convection-diffusion problems past curved obstacles (see [21]).
We first setwith which the main balance equation (228) becomesTo obtain a limiting PDE on the scale, we first formally expand (277) using , where we anticipate that will be scaled to be large, with and also . We thus rewrite (277) aswhere the error term(s) involve derivatives of of order and terms of order . We furthermore setso thatwhere we isolated dominant term in (281) on the scale . Setting and , multiplying (278) by , and using (279)–(281), we obtain the limiting PDEwhich applies over the quarter plane , , and we now view as a function of instead of . By rescaling and usingand lettingwe obtain from (282) the separable, parabolic PDEIncluding the exponential factor in (284) allows us to eliminate the last term in the left-hand side of (282), while the other factors (such as and ) are purely for convenience. We will show, by asymptotic matching between the and scales, that must be proportional to , and including the factor in (284) will lead to being asymptotically .
Next we examine the boundary condition in (18), which can be also written as , or, using the variables, asIn view of (276) and (279) we can also write (286) asorwhere we used also (280) and (284).
On the scale we have so as long as we conclude from (288) that must vanish along for all , and thus, in view of (284) so must ; henceThe case where corresponds to , and then the boundary condition will become more complicated. First we analyze the interior of where (290) holds.
To analyze (285) with (290), we first derive the behavior of as , by matching the expansions on the and scales. To this end we need, for region , the behavior of for with . This asymptotic limit lies in the asymptotic matching region between (249) (which applies for for region ) and (253), which applies for but with . Setting in the integral in (245) and shifting the contour toward we obtainHere we usedand approximated by a vertical contour. Evaluating the integral in (291) explicitly we conclude thatBy asymptotic matching the expansion on the scale must behave as the right side of (293), when with fixed. But , , and (283) shows thatThen comparing (293) with (276) (using (279) and (284)) we conclude thatNote that (295) is consistent with the boundary condition in (290) along . The exponential factors in (279) and (284) do not enter the matching condition, since and , and we can choose so that in the matching region.
To solve (285) subject to (295) it is useful to view the function as being an approximation to the delta function for , with the mass concentrated in the range . Then the right side of (295) corresponds to the dipole . Consider the problemIntroducing the Laplace transform(296) becomesand (298) implies that . The problem in (300) is a standard Green’s function problem with solutionwhere and are the Airy functions. Then the solution to (285) will beBy differentiating (302) with respect to , setting , and noting the Wronskian identity we find thatInverting the Laplace transform in (299) leads to in (170), so we have established the expansion on the scale. We can also easily verify that (295) is indeed satisfied, sinceHere we approximated the first integrand for using asymptotic properties of .
We next relate the constant in to ( in ), having expressed the expansions near the corner point in terms of on both the and scales. As we discussed previously, the caustic ray expansion in does not satisfy the boundary condition along , and different expansions must be constructed on the and scales, corresponding to and . For region , these expansions are given, respectively, in (111) and (109) (with (110)), up to the constant . The details of their construction are given in [10]. We will need to carefully estimate for . For region , (109) applies for and all (but breaks down as either or ). For and , (241) holds as then , and in (253) we gave the asymptotics for for and . However, we must derive yet another expansion on the scale with and . The expansion in (169) develops a nonuniformity as , since . Since by definition of the Airy roots we have , from the infinite series form in (170) we conclude thatand this holds for any fixed . Consider the scale and . Viewing in (277) now as a function of and and noting that corresponds to the limiting form of (277) becomesBut so that the general solution of (307) is the linear function , where and are arbitrary functions of . Note also that (307) applies asymptotically whether the exponential factors in (279) and (284) are included or excluded. The boundary condition in (287) can be applied on the scale and leads asymptotically to . This restricts the functions and in the linear solution to (307), byWe have thus shown that the expansion on the scale is given byHere we made proportional to , which we will determine shortly, and computed from (308). Now we require (309), for , to match to the scale result, for . But in view of (306) we see that the matching is possible provided thatThus we now have three expansions valid for , on the scales , , and with . The matching between the result and (309) follows by letting in (241) (corresponding to (253) for region ) and letting . The matching condition is satisfied ifin some intermediate limit where and . Simplifying (311) by using (283) we must show thatThe Airy roots are well known to satisfy , , so for we approximate the sum in (310) by an integral to getwhich verifies (312).
Now we match (309) for with (109) as . For we have as the zeroth term in the sum in (310) dominates. By expanding (109) for , after canceling some common factors the matching implies thatNow, By using (110) to compute and and noting that we see that the dependence on is exactly the same in the left- and right-hand sides of (314). Thus we have a relation between the constants and for region , and since we have derived the expression for in (41).
The analysis for region breaks down as , as then the expressions in (284), (293), (309), and (314) all become singular. We thus examine the transitional case , where . Then we certainly have and (241) applies for any . But now on parabolic scales, where with fixed, has the expansion in (254). We now define by settingsince the factors in (317) correspond to the same sequence of transformations we made for the analysis of the interior of , and will again satisfy the PDENote that in (317) we included the factor whereas in (284) we have the two factors and , as the product of these becomes, in region , . Including the factor will insure that is for , and the factor is purely for convenience. But for region we must reexamine the boundary condition in (288), which in terms of becomesBut now so that the limiting boundary condition from (319) becomesThen scaling , in terms of (321) becomesA matching condition for , as with , is obtained by comparing (317) to (254). From the definitions of and we have so that . Then alsoare all , while and , so in the matching region, where is small but , the Gaussian first term in the braces in (254) dominates the error function second term, and the matching will imply thatand in terms of this simplifies toHere we also used in region , so that .
The matching condition in (325) may be replaced by the condition and we analyzed such problems in [20] where it was found that the solution to (318) with (322) is given byThe right-hand side of (326) clearly satisfies (318), since this PDE is separable and any function of the form is a solution, for any function and any complex . We can also verify that (322) holds along , by using and some integration by parts in (326). To verify the matching condition in (325) we expand the integrand in (326) for , , and , using (190) to approximate the Airy function. We have andHere we also expanded the integrand for small , since by Watson’s lemma the main contribution comes from the range . Using (328) in (326) and noting that we obtain, for with ,The expression in (330) is a two-term asymptotic approximation in this limit, and the leading term verifies the matching condition in (325). The correction term in (330), which is proportional to , will asymptotically match to the term proportional to the error function in (254), as in the matching region and are , and
We proceed to determine . Now the approximation in (209) (with (326)) remains valid on the scale , since . Thus for can be approximated by (317) with replaced by and for the expansion of (326) is determined by the pole at , which is a simple pole if . Hence,For and , the result in (109) holds, and this remains finite as even if . In parameter region we have and then as , (109) becomesComparing (333) to (317) with (332), the exponential parts agree as for region , and we now haveThus (333) agrees with (317), with and , ifBut now and thus (335) leads to the result in (37).
We have now determined the values of , , , and for all possible regions of parameter space, thus establishing Propositions 1 and 2. We also have analyzed all of the relevant scales near the corner point of the state space and thus established all parts of Proposition 15. As a final step we briefly discuss the asymptotic matching between the ray expansion in and the expansion on the scale in (169). We only consider region and note that this matching will involve letting along parabolic scales with held fixed and letting in (169). We have already verified that along linear scales, where with () fixed, the ray expansion matches directly to (168). The matching of the and scale expansions will show that there are no “gaps” in the asymptotics, which would require the analysis of yet other scales.
We first expand in (170) for , and we note that, along parabolic scales with , corresponds to moving from the corner in region , while corresponds to moving into . By using (190) to approximate the integrand in (170) and shifting the contour to the right, to the range , we are led toHere we can view as being scaled to be and to be . The integrand in (336) has a saddle point whereand this occurs whenTaking then a standard saddle point estimate leads toBy using (339) in (169) we match the result to the ray expansion, which is , as . By separating the exponential factors from the algebraic ones, the matching will hold if bothWe thus proceed to evaluate and as along parabolic scales, where is fixed. In this limit we have and , and it is useful to rewrite (68) asThe function in (275) vanishes in this limit, in view of the factor , and we haveExpanding the Jacobian in (71) for and leads toBut from (342) we have so thatwhere we wrote the result in terms of rather than . From the first expression in (342) we have so thatNow so using (347) and (345) we can easily verify that (340) is satisfied.
To verify (341) we first note that , Then peeling off the linear part of near , (341) is equivalent to showing thatWe can rewrite in (66) aswhere we have partially expressed the right-hand side of (351) in terms of and . Next we expand (351) as a triple Taylor series about , , and . We haveBut the relation in (216) implies that the coefficient of in (352) vanishes and also thatBy (347), on the scale, is so that , and thus (352) may be further approximated by (using )with an error, which is not needed for the matching verification since . We also haveFrom (342) we have so we can refine the estimate above (345) toAdding the expressions in (354) and (355) and using the estimates in (347), (353), and (357), (351) becomes, when multiplied by ,Comparing (358) to (350) we find, after some simplification, that they are identical. The comparison is facilitated by separately comparing terms proportional to , , , and . This verifies the matching between the ray expansion and that on the scale. The Airy functions that arose on the scale disappear in the limit where . If we let with , then the expansion of is much different, and now we haveas the pole at determines the asymptotic behavior. We can show that then (169) with (359) agrees with the ray expansion in , that is, , as the latter is expanded for , for parameter region . The case where will be important in determining the transition layer expansion that applies where meets , which is along the curve . Then neither (339) nor (359) apply.
5. Asymptotic Expansions near State Space Boundaries
We discuss the four boundary segments, , , , and , of the state space rectangle, avoiding for now the corner points. From Figures 3–6 we see that will border and for parameter regions and and also border for region . As we previously discussed the ray expansion in remains valid near but breaks down near and . The analysis of (corresponding to the scales and ) is identical to that of the infinite capacity model, and we show in [10] how to construct appropriate boundary layer corrections; this leads to (88)–(95) in Proposition 8. The analysis near (with ) is also very similar to that in [10], except that if we must use the values of in (32) and (33). Thus both the ray expansion and the boundary layer expansion, which applies for , are rescaled by a constant. We thus obtain (114).
For parameter region(s) , meets the line , and there the expansions in (109)–(112) apply. Their derivation for is identical to that in [10], while for the other subcases we must simply multiply both the ray expansion and the boundary layer corrections by the appropriate , from among (35), (36), (37), and (41). Similarly, the boundary layer correction along is given by (96), with the appropriate value of . For this covers the entire range , but for only the range (with defined in item (vii) in Proposition 8), as in the range , meets .
Here we will only examine where meets the state space boundaries, and this occurs along for all regions , along with for , and all for , and along with for only. We thus proceed to construct boundary layer corrections to the ray expansion for these three boundary segments.
5.1. The Boundary Segment
We consider the scale and letThen the main balance equation (3), after dividing by , becomesTaking to be the leading order approximation to , dividing (361) by , and letting lead to the limiting equationBy examining the boundary condition in (6) we conclude that (362) holds also if , and thus
To determine and we asymptotically match (360) to the ray expansion. We thus expand as and compare the result to the large expansion of (360). By using Stirling’s formula we see that the matching is possible if, as ,Note that the exponential factor in (360) must be included for the matching to be possible. Thus we could set and then will be the limit of as , which we show below to be finite and nonzero.
From (68) we see that corresponds to and we set , where we view as a function of via the mapping in (68). Then we also set and from (68) find thatUsing (366) in (67) leads to a quadratic equation for , whose solution is given by (118). Also, replacing by in (66) leads to the expression in (119) for , which is an explicit function of . As , is finite and from (68) we obtainEvaluating the Jacobian in (71) along leads to in (120). Thus as we have
We have thus identified asUsing (366) and in (275) we see that (360) agrees with (117) as , ifand has the small behavior indicated in (364). Now, so that and using (366) leads toBut from (118) we see that so that the right-hand sides of (370) and (371) agree.
Using (68) we have, as ,
Integrating the above, noting that is an analytic function of , we conclude that and this verifies (364). We have thus shown that the asymptotic matching holds and also determined . This completes the derivation of (117)–(120).
For region , (117) holds for all , and then the expansion matches to the corner expansions that apply for () and (). But for region and , (117) holds only for , while (114) applies for . To connect these it is necessary to construct another expansion for . Using the variables and we conclude from (3) and (6) thatwhere we note that when , since . To determine we first infer its behaviors as by asymptotically matching (374) to the expansions that apply for and .
We expand the result in (114) as . We have so thatAfter some calculation we find thatFurthermore,and we note that, in view of Propositions 1 and 2,which holds in regions . Using (375)–(378) to infer the behavior of (114) as and comparing the result to (374) we can take and then, as ,
Next we expand (117) as . Since , (117) is singular in this limit, and we use to obtainwhere was defined in (125). To obtain (381) we also usedwhich follow from (118) and (120). Denoting the ratio of the left- and right-hand sides of (379) by , the matching conditions as may be written as
After some calculation we can show thatso that the error functionsatisfies the conditions in (383). To determine completely we need to match (374), with , to the expansion in the transition layer where and meet, which is given by (176). This matching has in (176) and in (374), with fixed. Noting that as we can show, using (178) and (184)–(187), that as and thus (385) determines for all .
5.2. The Boundary Segment
We consider , in ranges where meets the -axis, which can occur only for . The analysis for where (region ) or (regions and ) meet the -axis is very similar to that in [10] so we omit it. By examining (3), noting that this equation holds even if , on the scale with , , we find thatwhere will satisfyNoting that , we define by . A ray in reaches the -axis when so we also let . Then evaluating (197) along leads toso that the solution to (388) isWe proceed to determine the constant and function by asymptotically matching (387) to the ray expansion.
First we note that setting in (66) and replacing by lead to the expression in (99) for . As we also have, in view of (389),By integrating (391) we see that the ray expansion behaves for asHere we used (in ) and . Expanding in (390) by Stirling’s formula we see the matching is possible if andWe denote by the limit of as . Writing we simply replace by in the factors that are not singular and use Then becomes precisely the expression in (102). We have thus determined as and thus derived (98)–(102).
The approximation in (387) applies only for , and (96) applies for . But for a new approximation is needed, as vanishes as , due to the factor , since . Since we find from (102) thatBy implicit differentiation of (100) we find thatUsingand (397) we obtain from (396) the expression in (104), which will be used for asymptotic matching verifications.
We consider the scale , retaining . From the balance equations we can conclude that the expansion has the formwhere the last exponential factor follows from expanding about , and these are necessary to have a chance of matching to the expansion for . By comparing (399) to the behavior of (98) as we immediately conclude that so we set , and then
From (101) we find thatso that . Then from (99) we find thatand thus is as in (107). From (402) we also conclude thatBy further differentiation of (100) and (401) we find thatand thus
We next derive a matching condition for as . We first note from (97) that if we write thenFrom (406) we then obtainFrom (80) with we have from which we can show thatBy using the expression for in (41) we haveWith (409)–(411) we have obtained the behavior of (96) as and thus derived the matching condition
Thus (400) and (412) yield the behavior of as , but to determine this function for all we must use a third matching condition, to the transition layer in Proposition 18, which applies for . Thus we let in (191) and let in (399), in such a way that is fixed. Since this means thatremains also . From (86) we haveWe can expand in (191) as which is equivalent to expanding for and show that the exponential factors in (191) agree with those in (399), after is approximated by Stirling’s formula, so that Then using we compare algebraic factors in (399) and (191), which yieldswhereIn this limit and is fixed, so that (416) determines asWe have thus derived the result in (105). By using the asymptotic results in (192) and (193) (with replaced by ) we see immediately that (412) holds. The matching condition in (400) will also hold since by (417), (405), and (410). This completes our analysis of the scale .
5.3. The Boundary Segment
We consider the scale and . Now the analysis will be the same for any region , and the expansion we construct will hold everywhere except near the corner points and . By expanding (3) and the boundary condition in (5) along , we find that an asymptotic solution in this range is given bywhere is given in (128). Again, (420) must contain the factor in order to have a chance of matching to the ray expansion, and this factor determines the geometric factor in . We must only determine and by asymptotic matching.
As we have and in (68). Also,and from (68) we obtainwhere is obtained by solving the first equation in (68) with replaced by . Using (421) and (422) we conclude thatExpanding by Stirling’s formula and comparing the result to as we see that we can take and the matching will hold ifTo expand (70) as we note that in this limit and are finite, while vanishes and is singular, due to the factor in the denominator. We thus haveHere is the Jacobian in (71) evaluated along , and we also used the identity . From (67) we see thatso that with (422) we have, as ,Using (428) in (425) we can identify and then (420) becomes the same as (127).
Finally we note that as , (127) will asymptotically match to (168), the approximation valid for . For with , (168) can be approximated by (247). As we have , , , and . Then and thus, for , (127) reduces to (247). This completes the analysis of the scale .
6. Approximations near State Space Corners
We consider the four state space corners. The upper right corner was analyzed already in Section 4, since this was necessary to completely determine the ray expansion in . The analysis of the lower right corner is essentially the same as that for the infinite capacity model in [10]. For regions and the leading terms for near this corner are the same as for the infinite capacity model, while for regions and the analysis differs only through the multiplicative constant . Thus we omit the analysis of these two corners, focusing on the lower and upper left corners, and .
6.1. The Corner
For regions the analysis is very similar to that in [10], leading to (132)–(136) in Proposition 13. For region we setrecalling that . Dividing the main balance equation in (3) by , using (430), and letting denote the leading term approximation for , we obtain in the limit Note that (6) implies that (431) holds along , . Equation (431) has many different solutions, and anything of the form will be a solution provided that . Let us writeand we will show by asymptotic matching that only one value of is needed, and the matching will also determine the appropriate value. For regions the same argument was used in [10] to determine as , leading to (136). For region we can asymptotically match (430) to (117) (then with ), to (98) (then with ), or to the ray expansion (then with fixed). We discuss only the first two matchings, as the third will lead to the same conclusion.
For fixed and , (430) and (432) becomewhere the last formula holds in the matching region where but , and we wrote the expression in terms of . To expand (117) as , we observe from (118) that . Then comparing the geometric factors in , in (117) and (433), we must haveFrom (119) we find that as which implies that , consistent with (434). The matching also implies thatAfter some calculation we find from (120) thatand thusWe have thus derived (137), since for region .
The same conclusions follow by matching to (98), as then, for fixed and , (430) with (432) becomesWe expand (98) as . We have and from (100) find thatThen also and, using (101),We have, as ,By using (442) to infer the small behavior of and comparing the result to the exponential part of (439), we again conclude that is as in (434). To expand in (102) as we note that it is singular due to the factor Then after some calculation, using (440) and (441) we find that has the expansion in (103) as . Expanding (98) as and comparing the result to (439), noting that , regain the expression in (438) for . We can also easily verify that (137) matches to the ray expansion, by expanding the latter for along lines of constant slope .
6.2. The Corner
Now we use the variables and , withNear this corner only the balance equations in (3), (5), (6), and (7) apply. Using (444) in (3) we obtainLetting be the limiting form of as , (445) leads towhose most general solution isEquations (5)–(7) provide, in the limit , no additional information. To determine in (447) we use asymptotic matching to the boundary layer expansion in (127), which applies for and .
As we have , and then is as in (131). Also, from (68) with replaced by we find that and then as . Using the above in (127) we find that for we haveBy comparing (449) to the large expansion of (444), with (447), we can takeand then is determined asWe have thus established the result in (130).
We conclude by showing that (130), for and fixed , matches asymptotically to (117), for and fixed . In this limit, (117) is singular due to the factor . First we note from (118) that and from (119) we get and thusThus,After some calculation we find from (120) thatand alsoWith (454)–(456) the expansion of (117) agrees precisely with the large behavior of (444), with (447), (450), and (451). This completes the matching verification.
7. Approximations near Transition Layers
We analyze the vicinities of the curves where the regions , , and meet. From Figures 3–6 we see that, for , meets along the curve . For regions , also meets , while and meet along . For region , and meet along . The analysis for , with the scaling , is carried out in [10] and we omit it here. For the analysis is exactly as in [10], while for and the and ray expansions must be multiplied by the appropriate constants and , but the analysis is otherwise unchanged. We thus obtain the result in (188).
7.1. Transition Layer near , Region
This layer arises only for parameter region . We set and use as the variables. Let us write in the ray expansion in (79) aswhereThe scaling corresponds to rays that have and more precisely . We thus set
In view of (75) and (86) we havewhich relates the two curves and . From the definition of in (81) we then haveso that
Consider the balance equation (17) on the scale , with . An asymptotic solution is given by where we can replace by , so the expansion is in terms of and . We can also view as being a function of the ray variables and . But then the product will satisfy the transport equation in (198). Since is a particular solution, must be constant along a ray and thus a function of but not , so in view of (462) we writeWe thus write the expansion in the transition layer aswhere we used . If (465) is to asymptotically match to the ray expansion we must haveUsing the fact that we have
Next we examine the ray expansion in near the curve . This will yield a matching condition for as . We can easily establish the following continuity conditions between and across where we recall that corresponds to the ray with , . To this end we note thatwhere we expressed in terms of and using (68). Along we have so that from (469) we obtainand henceFrom the relation between and in (67) we haveand thusIt also follows from (470) thatEliminating in (68) leads to
By implicit differentiation of (475) we obtainand then using (473) in (474) leads toCombining (468) with (477) we have
As , ,The function vanishes as , in view of the factor , and we haveCombining (479) and (480) leads to, for ,Using the expression in (41) for and the continuity conditions in (468) and (477), we compare (465) to the ray expansion as to conclude thatThus (466) and (482) give the behaviors of as . But to determine completely we must use asymptotic matching to the corner approximation in (169), which applies on the scale. We thus expand (465) for and asymptotically match this to (169), expanding the latter for , but with . In this limit we haveRecalling that we haveand thenApart from the exponential factoras the expression in (465) becomesHere we note that is singular as , in view of the factor , and in this limit The limit of (169) as , using (483), apart from some exponential factors, which will automatically match those from (486), is given byComparing (487) with (489) determines asWith (490), (465) becomes the same as (191), so we have established Proposition 8. Note also that (490) is consistent with (466), since for the asymptotics of the contour integral are determined by the pole at . For a standard saddle point calculation shows that (490) is asymptotic to which is consistent with (482).
7.2. Transition Layer near , Region
We take . Now the curves , , and are all close to each other, coinciding if . Most of the analysis closely parallels that in , so we include here fewer of the details. We again use the variables and to find that in the transition layerThe factors in (492) that precede come from , (now given by (37)), , , and , except we exclude from the factor , which vanishes if and . Instead we include an extra factor of , and then , where comes from in (37). Also, in (492) In region we have and , and also Thus apart from the factors , (492) is just the ray expansion expanded near (or ), divided by . Thus (492) matches to the ray expansion ifBut the left side of (495), in view of (461) and (462), becomesNow so that (495) and (496) lead to
Next we match (492) to the ray expansion and thus infer the behavior of as . Now . First evaluating at and then letting we getThe estimate in (478) still holds so if is to agree, for , with the large asymptotics of (492), we must haveWe thus have the matching conditionTo determine completely we need to match (492) to the corner layer expansion in (173). Using in the integral we expand (173) for with . The result contains some exponential factors and some algebraic ones, with the latter being
Now we evaluate the algebraic part of (492), for . From (460), since , we see that is the same as , which is independent of . We also have so that (498) becomes singular as . We haveThe result in (485) still applies, now simplifying toWith (503) and (504), the algebraic part of (492) becomesComparing (502) to (505) determines asWe have thus established the result in (194). By asymptotically expanding the contour integral in (506) in the limits we can easily verify that (497) and (500) are satisfied.
7.3. Transition Layer near , Region
We consider the curve that separates from . We use the scalingand expand the joint distribution asThe exponential factor in (508) corresponds to the expansions of about . If (508) as is to match to the ray expansion we must haveWe use (508) in the main balance equation (3) and after a lengthy calculation we find that satisfies the parabolic PDEfor and . In (510), is understood to be evaluated at . We can write the curve in parametric form, aswith the latter equation corresponding to (178). Since we note thatand define from
We can view as being a function of either or . In terms of we haveChanging variables from to the PDE in (510) becomesand note that corresponds to , as .
Next we assume that will be a function of a single “similarity” variable, which we call and with which (515) becomes the ordinary differential equationwhere Such a similarity solution is possible if satisfies the nonlinear ODESetting , the Bernoulli equation in (518) becomes the linear equationSolving (519) subject to leads to the expression in (184). With (518), (516) becomesand (509) implies that , and thusWe have thus derived (176).
With (521) substituted for in (508), we can show that as , (508) asymptotically matches to the ray expansion. As from (521) we haveWe can easily establish the continuity conditionsand, after a lengthy calculation, show thatThus expanding the ray expansion, , as and comparing this to the large expansion of (508), using (523) and (524), lead toBut from Proposition 2 we see that, for , The curve corresponds to the ray , and then . From (70) we see that becomes singular as , due to the factor . This singularity precisely matches that in the right-hand side of (525), and yet another lengthy calculation shows that which establishes the matching.
We note that the PDE (515) contains many solutions other than the similarity solution. An initial condition as can be obtained by asymptotically matching (508) to the corner layer valid on the scale. This will determine the solution to (515) uniquely, but the matching will again lead to the conclusion that is given by (521). Below we only briefly verify that the matching holds. Near the corner the curve can be approximated by the straight line , which corresponds towhere was defined in Proposition 20. Thus we must show that (508) for is the same as (251). The exponential parts agree automatically and , so we must only show that, for ,But From (184)–(187) we obtain, as or ,since , as approaches in (252), which verifies (529). This completes the analysis of the transition from to .
8. Numerical Studies and Discussion
Next we show that our asymptotic results can also be used to accurately estimate the probabilities in ranges where there is little mass. We will consider which is the (very unlikely) situation where no primary spaces are occupied but all of the secondary spaces are full. To estimate we must use the expansion that is valid for and , and this corresponds to (130). Thus we set in (130) and note that has the different expansions in Proposition 1, according to regions in parameter space. In Table 1 we take , and use the expression for that holds in parameter region . We see that the exact and asymptotic results agree to three decimal places even if is as small as 2. In Table 2 we have and , and is again computed from (30). Now the agreement is not quite as good as in Table 1, and we typically have errors of about 10%. This is probably due to the fact that numerically , which exceeds one only slightly, so it may be preferable to use (29) to approximate . In Tables 3 and 4 we take, respectively, , and , . Table 3 corresponds to region and Table 4 to region . The agreement is certainly better for region , as again the sum is further away from the critical value of one. Here we used for region(s) .
Tables 1–4 show that the very small values of are well predicted by the asymptotic formula(s).
To summarize, we have done a rather thorough asymptotic analysis for this storage allocation model. We have shown that as long as () the effects of the finiteness of the secondary storage capacity occur only for and for those state space regions that border . However, for () the entire state space is affected by the finite capacity, as then there are not enough storage spaces to satisfy the demand for them. For it proves useful to consider the numbers, and , of empty primary and secondary spaces, and then the steady state distribution has for the limiting form in (50). Even though the marginal distributions of and the sum are particularly simple, the joint distribution of is quite complicated, as the analysis involves many different ranges of the parameter space (cf. Figure 2) and the state space (cf. Figures 3–6). In some ranges special functions such as Airy and parabolic cylinder functions play a key role.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work is partially supported by a Faculty Development Grant from Columbia College Chicago.