Research Article | Open Access
John A. Morrison, Charles Knessl, "Asymptotic Analysis of a Loss Model with Trunk Reservation I: Trunks Reserved for Fast Traffic", International Journal of Stochastic Analysis, vol. 2008, Article ID 415692, 34 pages, 2008. https://doi.org/10.1155/2008/415692
Asymptotic Analysis of a Loss Model with Trunk Reservation I: Trunks Reserved for Fast Traffic
We consider a model for a single link in a circuit-switched network. The link has circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the links, are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that is large and , the arrival rate of primary traffic is , while that of secondary traffic is smaller, of the order . The holding times of the primary calls are assumed to be exponentially distributed with unit mean. Those of the secondary calls are exponentially distributed with a large mean, that is, . Thus, the primary calls have fast arrivals and fast service, compared to the secondary calls. The loads for both traffic types are comparable , and we assume that the system is “critically loaded”; that is, the system's capacity is approximately equal to the total load. We analyze asymptotically the steady state probability that (resp., ) circuits are occupied by primary (resp., secondary) calls. In particular, we obtain two-term asymptotic approximations to the blocking probabilities for both traffic types.
A classic model in teletraffic is the Erlang loss model. Here, we have servers (or circuits), and customers (telephone calls) arrive as a Poisson process with rate parameter . The arriving customer takes one of the circuits if one is available, and if they are all occupied then the call is blocked and lost. When occupying a circuit, the customer has an exponentially distributed holding time whose mean we take as the unit of time. It is well known that the steady state probability that circuits are occupied is the truncated Poisson distribution, that is, , , with . This model dates back to circa 1918 . When , we obtain the steady state blocking probability. The transient probability distribution is much more complicated, but it can be computed in terms of special functions (see ).
Over the years, many generalizations of the basic model have been analyzed, including networks of such loss models (see [3, 4]). One important extension is that of trunk reservation, which is fundamental in the analysis of circuit-switched communication networks. Here, we consider a model with circuits that are used by the two types of customers (or offered calls). We refer to these as primary (or high-priority) calls and secondary (or low-priority) calls. They arrive as Poisson arrival streams with respective rates and . Of the circuits, are reserved for primary calls. Thus, if a high-priority call arrives, it is blocked if all circuits are busy, while a low-priority call is blocked if at least circuits are busy. All calls are assumed to have independent and exponentially distributed holding times, with respective means and . The total load on the system is . If this exceeds , then typically all the circuits are busy (an overloaded link), while if , typically some circuits are free (an underloaded link). An interesting situation is when is large and ; we refer to this as “critical loading.”
Previous work on this and related models includes Mitra and Gibbens  who considered the asymptotic regime with , , and (thus, secondary calls are less frequent than primary calls). We thus have ; so this is an example of critical loading. They analyzed a single link and used their results to obtain approximations for more complicated loss networks with a distributed, state-dependent, dynamic routing strategy. Related work appears in [6, 7], and optimization and control policies for such problems were analyzed by Hunt and Laws .
Of fundamental importance in this model is the probability (resp., ) that a primary (resp., secondary) call is blocked and lost in the steady state. Roberts [9, 10] obtained approximations to these blocking probabilities, which are based on a certain recursion which is exact for special cases of the model parameters, but not for all cases. Morrison  investigated this model for and , and obtained the blocking probabilities as asymptotic series in powers of . This led to a better understanding of the asymptotic validity of Roberts' approximation(s). However, the coefficients in the asymptotic series in  were not explicit, as their calculation still involves recursively solving an infinite system of differential equations. But, if it is further assumed that is small, the blocking probabilities were obtained more explicitly in terms of parabolic cylinder functions. Also, if rather than , explicit results are obtained without the small assumption.
In , we analyzed the case , with and with the arrival rates and both . Expressions for the blocking probabilities were obtained for the overloaded and underloaded cases. In the first case, both blocking probabilities remain as , while in the second case they are exponentially small. In this paper, we investigate the case of critical loading, where again but now with . We will also assume that (secondary calls are less frequent than primary ones) but now with ; that is, secondary calls have large holding times. Thus, primary calls have faster arrivals and faster service. Note that the loads due to the primary and secondary calls remain asymptotically comparable with this scaling. In this asymptotic regime, we are able to obtain explicit analytic expressions for the first two terms in the expansions in powers of for the blocking probabilities, which involve readily evaluated definite integrals.
We are currently investigating situations where the secondary calls are the ones with fast arrivals and service . Here, the asymptotic structure of the problem turns out to be quite different.
We comment that the basic problem to be solved is a two-dimensional difference equation (cf. (2.1)), with discontinuities of the coefficient functions at various boundaries and an interface. Such a problem appears to be very difficult or impossible to solve exactly. From a numerical point of view, the problem corresponds to solving roughly linear equations. A good general method such as Gaussian elimination has computational complexity or . Some methods that use the sparseness of the system and some iteration procedures may improve this to or . The purpose of our asymptotic analysis is to obtain reasonable approximations whose numerical evaluation has computational complexity that is independent of , and also to obtain explicit formulas that show the dependence of the stationary distribution and blocking probabilities on the model parameters (i.e., , , and the arrival and service rates).
The paper is organized as follows. In Section 2, the problem is stated more precisely and the basic equations are obtained. Here, we summarize our main results, which are derived in detail in Sections 3 and 5. In Section 3, we obtain the leading terms for the blocking probabilities. In Section 4, we relate the present results to the ones in [11, 12] using asymptotic matching. The first-order correction terms to the blocking probabilities are derived in Section 5, while in Section 6 we present some numerical studies to assess the accuracy of the asymptotics.
2. Statement of the Problem and Summary of Results
We denote by the number of servers serving high-priority customers, and by the number of servers serving low-priority ones. The total number of servers (circuits) is of which are reserved for the high-priority customers. Thus, if , a newly arriving high-priority customer (call) is lost; if , then a newly arriving low-priority call is lost. The high- and low-priority customers arrive as independent Poisson processes, with respective rates and . The service times are exponentially distributed with respective means and . Thus, the unit of time is taken as the service rate of the high-priority customers.
We denote the steady state joint distribution of the numbers of servers used by the two priority classes by . We let be the indicator function on the event . Then, from the description of the model, we obtain the following balance equation:
This applies over the domain Thus, we may view the problem as solving a second-order difference equation in two variables, over the triangle and the oblique strip , with the two subdomains separated by the “interface” . There are also boundary conditions inherent in (2.1), along , , , and . The normalization condition is
Of particular interest are the blocking probabilities for the high-priority customers, defined by and for the low-priority ones, defined by Note that we clearly have .
We analyze the problem in the asymptotic limit where
We furthermore assume that the arrival rate of high-priority customers is large, of the same magnitude as , and then scale the other rate parameters as
Thus, the arrival rate of low-priority customers is large, but only of the order . The service times of these customers however are also large, and the total load due to low-priority customers is . Also, we have so that the total load due to all customers is roughly equal to the capacity of the system. Hence, this asymptotic limit may certainly be considered as “heavy traffic” or “critical loading.”
Once we input the model parameters , , , , and , we can compute , , and from (2.7). For some of our numerical studies, it is desirable to fix , , , , and and then vary the original rate parameters, which are computed by inverting (2.7) using
and and are obtained from (2.7) once is known.
We next scale and as
with and and are taken as . We consider the scaled state space with . Then, since is large and is , the domain in the plane is the triangle ; . The scaling (2.9) corresponds to a small neighborhood of the point
However, this is where most of the probability mass accumulates in this asymptotic limit, and the analysis of this range is sufficient to obtain the blocking probabilities . We will obtain these as asymptotic series in powers of .
Note that in this asymptotic scaling, the indicator functions , , and may be replaced by one, since these correspond to boundaries that are far from the point in (2.11). However, the interface and the boundary are evident in (2.12) and will play a large part in the analysis. In Section 3, we will analyze (2.12), and then also consider a second scale where is large and negative. This second scale will lead to a diffusion equation in two variables.
The main results are as follows. For and , we obtain the approximation , with
where The correction term in (2.13) takes the form for and the form for , where For the latter function, we have , where is given by (5.5) in terms of and . Then, is given by (5.66)–(5.68) (with and defined in (5.6) and (5.56)), and the constant in (5.66) can be obtained by using (5.78) in (5.83).
On the scale, where and , we find that
where and is as above.
We note that the numerical evaluation of the leading order asymptotic results involves only the integrals in (2.15) and (2.17). The correction terms involve numerically evaluating some double or triple integrals (cf. (5.78)), but the computational complexity of evaluating the asymptotic results is independent of (or ).
3. Asymptotic Analysis: Leading Terms
We consider (2.12) and assume that for the probabilities have the expansion
In this section, we focus on the leading term, but its calculation will necessitate that we also analyze the problem for . This correction term is calculated completely in Section 5. We will also need to couple the analysis of the scale to that where is large and negative, with .
and this applies for all . This is a simple difference equation with a boundary condition at . Its most general solution is
where is to be determined.
whose solution is Here, is not yet determined. For , the terms in (2.12) yield
which simplifies to and hence
Note that this still corresponds to a local approximation near the point in (2.11). In terms of , we let
and (2.12), upon multiplying by , becomes
We note that corresponds to , and that for the indicator functions in (2.12) can all be replaced by one.
We assume that has an expansion in the form
Using (3.14) in (3.13), we obtain to leading order the PDE This is a parabolic PDE whose solution is facilitated by the change of variables Using (3.16), (3.15) becomes The most general solution, that decays exponentially as , is
and hence We will determine shortly.
We observe that on the scale, with , expansion (3.14) becomes
and, from (3.10) that It follows that which is a boundary condition for the PDE (3.15) along . In terms of , this becomes , but this holds automatically (for any ) in view of (3.18). To determine , we must analyze the correction term in (3.14).
Switching to the variables and then using (3.18), we get
Integrating (3.25) with respect to yields
Setting in (3.26), we obtain
We will show that (3.27) leads to a differential equation for . But, we must first consider . In view of (3.20), this term arises as a part of the term in the expansion on the scale. We therefore return to (3.1) and (2.12).
This equation may be written as a perfect derivative as Integrating once and requiring to vanish as yield
and hence where is a constant, which will be fixed by normalization.
which applies for and is independent of . To calculate correction terms to (3.46) and (3.47), we would need to find in (3.5) and (3.10), and solve (3.24) for . This ultimately involves calculating the and terms in (3.1) and the term in (3.14); this is done in Section 5.
We next calculate the blocking probabilities in (2.4) and (2.5). Evaluating these sums requires the expansion on the scale. Again, approximating sums by integrals and using the scaling (2.9), we obtain
4. Consistency with Previous Results
where We show that (4.1) matches asymptotically to (3.48), in the limit where . When , we can simplify the integrals in both the numerator and denominator in (3.49), as the factor has the effect of freezing the remaining integrands at . Thus, by the Laplace method, we obtain
In , Knessl and Morrison analyzed the model in the limit with and . The total load is thus and the cases (resp., ) correspond to an underloaded (resp., overloaded) link. The case of critical loading was not considered in .
For the matching, we must thus let with , and . The results in  for and were as follows:
where In view of (4.4), we have, in the matching region, , , , and
Hence, and (4.5) becomes We show that (4.9) agrees with (3.48)-(3.49) when the latter is expanded for . In (3.49), the main contribution to the integral in the denominator comes from , and in the numerator from , where . Thus, we can approximate by everywhere, and obtain
Next, we consider the overloaded case in , for which we obtained
where and is the solution of
We note that by using (4.12) in (4.13), is a particular root of a polynomial. For the matching, we will take in (4.11) and in (3.48)-(3.49). As , we will have and this will allow us to simplify (4.11). Let us set
with which , and thus
We expand (3.49) for . Scaling , we obtain
For , we use the asymptotic approximation
This completes the matching verifications.
5. Correction Terms
we obtain from (3.13) so that satisfies
where are the operators
Before analyzing (5.3), we obtain a more complete description of . We recall that is known completely, in view of (3.18), (3.43), and (3.45). But, we computed only partially as the combination in (3.26). We integrate (3.26) to get
where We observe that satisfies
and decays as . We also have
The function will be determined shortly (actually, not very shortly, but only after a lengthy calculation).
We evaluate the right side of (5.3) more explicitly. Some terms are expressible as derivatives with respect to , while the ones involving derivatives in may be evaluated using (3.18) and (5.5). Then, (5.3) becomes
From (3.18), we have and by direct calculation With the above, we integrate (5.9) to get where We will show that the calculation of will require only that we evaluate (5.12) along . However, we must first reconsider the scale and analyze at least partly the term in (3.1) (i.e., the coefficient of in the series).
We then rewrite (5.14) as