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Advances in Operations Research
Volume 2013 (2013), Article ID 680539, 21 pages
On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits
1Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA
2Department of Mathematics and Computer Science, QCC, The City University of New York, 222-05 56th Avenue, Bayside, NY 11364, USA
Received 28 January 2013; Accepted 8 April 2013
Academic Editor: Khosrow Moshirvaziri
Copyright © 2013 Charles Knessl and Haishen Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotes of its capacity to the first queue and the remaining to the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network. Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable.
We consider a nonsymmetric version of the longer queue model. Here there are two parallel queues, each fed by a Poisson arrival stream. There is but a single server who tends to the longer of the two queues. If the number of customers in each queue is the same, then the server devotes of its capacity to the first queue and to the second queue, with . We let () denote the number of customers in the first (second) queue, the two arrival rates are and , and the server works at rate . Note that the total number of customers, , in the two-node network behaves as the standard model, so in the steady state we have the geometric distribution , where , assuming the stability condition . The “symmetric case” corresponds to (thus ) and , and this was analyzed in detail by Flatto .
Such models were proposed by Zheng and Zipkin  to study problems in inventory control. In  finite capacities were assumed in the two queues, and the authors studied numerically the steady state probabilities , in terms of the capacity size and also for different service disciplines, such as the longer queue (LQ) discipline here, and also the first-come-first-served discipline.
In  the author used two-dimensional generating functions and analyticity arguments and obtained explicit expressions for , in the symmetric case, as contour integrals. Then asymptotic results were derived for the joint distribution , as and/or becomes large, and also the marginal tails and various conditional limit laws were obtained. In this paper we generalize some of the results of  to the nonsymmetric model, and we will show that now many of the asymptotic results become quite different. As in  we assume that the model is preemptive, so if (and thus the server works on both queues) and a new arrival occurs to the first queue, then the server switches immediately all its capacity to the first queue. The more difficult nonpreemptive version of the LQ model was studied by Cohen , in the case of Poisson arrivals and general service times. The problem is reduced in , using generating functions, to a functional equation which is furthermore converted to a Riemann-Hilbert boundary value problem.
The present problem corresponds to a random walk in the quarter plane (as ), and other examples of such problems include shortest queue (SQ) problems [4–6], fork-join models [7, 8], two coupled processors with generalized processor sharing [9–11], and two coupled tandem queues . General techniques for solving such problems are discussed in [13–15] and they involve functional equations, analyticity arguments, and singular integral equation methods .
For the present model the analyticity arguments are fairly simple, and we focus mostly on the asymptotic properties of the solution. We will show that these asymptotics are quite different from those of the symmetric model in . After obtaining exact integral representations for , and also the marginal probabilities and , we asymptotically evaluate these integrals for and/or large. We use standard techniques, such as the Laplace method, saddle point method, singular analysis, and the Euler-MacLaurin formula. Good general references on the asymptotic evaluation of integrals and sums are the books [17–21].
In addition to this model being interesting on its own, many variants of shortest queue problems, such as ones with multiple servers and finite capacities, can be asymptotically reduced to LQ models of the type considered here (see [22, 23]). For example, in  we showed that the finite capacity version of the standard symmetric SQ model (analyzed in [4, 5]), where and is the capacity, asymptotically reduces to the symmetric LQ model in , if we consider the process , which measures the number of spots available in the two waiting rooms. Then having a thorough understanding of the nonsymmetric LQ model and its asymptotics will allow us to analyze, at least in some asymptotic limits, nonsymmetric variants of SQ models.
The remainder of this paper is organized as follows. In Section 2 we summarize all of the main results, both exact and asymptotic. They are listed in Theorems 1–5, and some discussion/interpretation appears following each theorem. In Section 3 we briefly derive the exact expressions for ; in Section 4 we derive asymptotic properties of for large. In Section 5 we derive light traffic (where ) and heavy traffic (where ) results.
2. Problem Statement and Summary of Results
We let be the numbers of customers in the two parallel queues, and let be the joint queue length distribution in the steady state. The two arrival rates are and , the exponential server works at rate , and . If the server works on the first (second) queue, but if the server works at rate on the first queue and rate on the second, with . The symmetric case corresponds to and . We henceforth assume the stability condition .
In Figure 1 we sketch the transition rates for the random walk , which illustrates the discontinuity along the “interface” . The main balance equations are We also have the following three interface equations: the two boundary conditions and the corner conditions The normalization is Note that the elementary difference equations (6) and (7) may be solved immediately to obtain and , up to multiplicative constants. In Section 3 we analyze (1)–(9) to obtain in the forms of contour integrals, which we summarize below.
Theorem 1. For , the steady state distribution is as follows: (i): where the integral is over a small loop about , (ii): (iii):
We next evaluate in various asymptotic limits, to gain more insight into the structure of the joint distribution. Writing to emphasize the dependence on the model parameters, we clearly have the symmetry relation Thus it is sufficient for the asymptotics to assume that and we also note that the expressions in Theorem 1 are consistent with (22).
We will show that the asymptotics are quite different whether , or . In Figure 2 we sketch the curve in the parameter plane, for . Note that the curve may also be written as and passes through the points and . We first give results for when , and note that this includes the symmetric case of Flatto .
Theorem 2. For and one lets be the queue length ratio. The following asymptotic expressions then hold:(i), (ii) with , where and can be computed from (12)–(14).(iii) with (thus ) (iv) with (v)(vi) with , (vii) with (thus ) (viii) with and can be computed from (14), (16), and (17).(ix),
In Theorem 2 we listed the expansions of in order of decreasing , from in (25) to in (46). In the symmetric case and , and then and Theorem 2 is consistent with the results of Flatto in  (there the cases , and , were not considered, and the limits where were not treated in as much detail). Note that (34) and (38) correspond to “product form” approximations to . The expressions in (25) and (46) are actually exact when and , respectively, in view of our comments below (9). Our analysis in Section 4 will also indicate how to compute higher order correction terms in the various asymptotic series.
Next we take . Now some of the asymptotic results for will be very different, while in other ranges of they will be similar to those in Theorem 2. We emphasize below the formulas that are different.
Theorem 3. For and one has the following asymptotic expansions:(i) with and (ii)(iii) with (including the limit ), (iv) with (thus ) and is the parabolic cylinder function of order .(v) For and , (25) applies. For with , (27)–(30) hold. For with , (42)–(45) hold. For and , (46) applies.
Thus when the asymptotics of are different in eight ranges of , but the final result in four of these is the same as the case . Note however that if (27) applies for all , so the transition for is now absent. Also, if , (42) applies for rather than . In (57), is computed by setting , cf. (49), in (16), and we also note that .
The results in (49)–(60) are very different in form from the symmetric case asymptotics in . For example, the diagonal probabilities in (52) contain the additional algebraic factor . In (57) the parabolic cylinder function can be computed, for example, from where is a vertical contour in the complex -plane with , and we have for the correction term in (57). Also, and for we can express in terms of a Hermite polynomial. In (57) we give a two-term asymptotic approximation in the transition range . Also, it may be shown that the leading term in (57) is just the limiting form of (42), as . The expression in (55) applies both to with , and to with . In the latter case we can approximate the factor by . The parabolic cylinder function has the asymptotic behaviors so as in (57) (corresponding to ) the correction term begins to dominate the leading term (which occurs for or ). Then we can show that the expression in (57) for agrees with (55) for , and hence the two expansions asymptotically match. Note also that the scaling in (57) has .
Next we study the transition range in parameter space. This will lead to a new set of asymptotic results which will show, for example, how the formula in (52) for changes to the purely geometric approximation in (35). To quantify the closeness to the curve in Figure 2 we write where and can have either sign. If we are exactly on the transition curve. For small , in certain ranges of , the results in Theorems 2 and 3 still apply. For example, if and or and Theorems 2 and 3 agree, and then no transition range result is needed. As , in (26), and then (27) will hold for all in the transition case. Thus for the transition case will require a new asymptotic result only if . For the asymptotic result in (42) will apply for , where now, since , (cf. (37) and (54)). Then if , we will need different asymptotic results only when , including and , where Note that is the limit of both and , if we replace by , as (37) leads to (setting , ) since corresponds to . Similarly, (54) leads to in this limit.
Since we will now have the state variables and large, and also small, it is necessary to relate these. In Theorem 4 we summarize the transition case results, scaling and in terms of .
Theorem 4. For and with , one has the following, where and .(i), (ii)(iii)(iv), (thus ) and is the parabolic cylinder function of order (see (61)).(v), , and is defined by the contour integral where is to the right of all singularities of the integrand, including the pole at if .
If then and the term with the integral in (66) is absent. In Theorem 4 we wrote most results in terms of and . As it can be shown that (66) asymptotically matches to (35), and as , (66) matches to (52). Similarly as (68) matches to (27) while as (68) matches to (48). Some of these asymptotic matching relations are further discussed in Section 4. In (70), depends on and also on the ratio , in view of (71). The results in Theorem 4 show that for it is natural to scale , and then either independently, or . The expression in (74) is similar to, but somewhat more complicated than, that in (72). The function in (76) reduces to in the limit , and thus (72) may be viewed as a special case of (74)–(76). Note also that the results in Theorem 4 can be rewritten without introducing and in (63), as for example and is proportional to , and may be written in terms of the original parameters .
Next we consider some different asymptotic limits, those of “light” and “heavy” traffic. Light traffic corresponds to infrequent arrivals, where . Heavy traffic corresponds to nearly unstable systems, where . It turns out that the present model has two possible heavy traffic limits. In the first, which we call HTL1, we have with a fixed . Then most of the probability mass will occur in the range where and are large, but with . More precisely, if then and must be scaled to be , but with the difference fixed. In the second heavy traffic limit (HTL2) we again set but now let , with . Now the probability mass will become more spread out, with appreciable mass anywhere in the range , where and . The light and heavy traffic results are summarized below as Theorem 5.
Light traffic: For one has
Heavy traffic limit with ,
Heavy traffic limit with , , Here is a vertical contour in the -plane, which lies to the right of all singularities.
The expression in (78) applies for , , and , and in the light traffic limit the discontinuity of along the diagonal will appear only in the higher order terms. In HTL1, (79)–(81) show a piecewise geometric distribution in the variable, and an exponential density in . For HTL2, writing (82) as we can easily show that so that to leading order the probability mass concentrates where with . From (84) we have but the total mass along the main diagonal is , which is smaller than the mass in (82). Then also with total mass , which is comparable to that along the main diagonal. The diagonals with have mass , which is smaller still. The integrands in (82), (84), and (86) have branch points at , are analytic at , and may have poles at , if .
This completes our summarization of the exact and asymptotic results. Despite the seeming complexity and the many separate cases, all the results follow from fairly standard asymptotic evaluations of the integrals in Theorems 1, as we will show in Sections 4 and 5.
3. The Exact Solution
If we define then from (96) and (88)–(90) we find that Using (93) with , (94) with and (95) with , we obtain for the simpler equation so that and then by (92), and thus . Then from (96) we have so that follows a geometric distribution with parameter , and hence behaves as the standard model.
The coefficient of in (93) has roots at while that of in (94) has roots at By using (95) to express in terms of and , eliminating in the right sides of (93) and (94), and requiring that be analytic at and that be analytic at , we obtain two equations for and : Solving the algebraic system in (98) and (99), and then using the result in (95) to compute , we hence obtain explicitly the right sides of (93) and (94). In particular, where is as in (12)–(14). Comparing (100) to (88) we conclude that for . Setting and inverting the generating function in (101) leads to (10) for . Similarly, inverting the double generating function for in (89) leads to (15), and we note that (10) and (15) are consistent with the symmetry . Finally, in (19) is obtained by inverting the generating function . We have thus established Theorem 1.
4. Asymptotics of the Joint Distribution
We derive Theorems 2–4 by expanding asymptotically the integrals in Theorem 1. We will use a combination of the saddle point method and singularity analysis. Good general references on techniques for asymptotically evaluating integrals can be found in [17–21].
We need to understand the singularities of the integrands in (10), (15), and (19). There are clearly branch points where and , with Since we have and for the branch point at is farther from the origin than the one at . In fact, will never play a role in the asymptotics. The integrands are also singular at , where (10) has a pole of order , and (15) has a pole of order . The only possible other singular points are at the zeros of . We can easily verify that is a simple zero of all four functions , , , and , so all the integrands are analytic at . In the appendix we study in detail the algebraic equation , and show that the only possible zero is at but only if (if then has a branch point at ). If then the two branch points are the only singularities of the functions , , and in (12), (16), and (19). If , which is clearly true in the symmetric case, then is a simple pole of these functions (since ). In view of (105) and (106), we have , if the pole is present and the stability condition holds.
Consider first (19). If the pole at determines the asymptotic behavior, as it is closer to than the branch point at . Hence as