A Queueing System with Queue Length Dependent Service Times, with Applications to Cell Discarding in ATM Networks

A Queueing System with Queue Length Dependent Service Times, with Applications to Cell Discarding in ATM Networks by Doo Il Choi, Charles Knessl and Charles Tier University of Illinois at Chicago 85 South Morgan St. Chicago, IL 667 e-mail: dichoi@hit.halla.ac.kr; knessl@uic.edu; tier@uic.edu Abstract A queueing system (M/G,G // ) is considered in which the service time of a customer entering service depends on whether the queue length,, is above or below a threshold. The arrival process is Poisson and the general service times and depend on whether the queue length at the time service is initiated is or, respectively. Balance equations are given for the stationary probabilities of the Markov process where is the remaining service time of the customer currently in service. Exact solutions for the stationary probabilities are constructed for both infinite and finite capacity systems. Asymptotic approximations of the solutions are given, which yield simple formulas for performance measures such as loss rates and tail probabilities. The numerical accuracy of the asymptotic results is tested. AMS Classification: 6K5, 4E5 Keywords: queueing systems, asymptotic methods, ATM networks Introduction. Background Queueing systems arise in a wide variety of applications such as computer systems and communication networks. A queueing system is a mathematical model to characterize the system, in which the arrivals and the service of customers (users, packets or cells) occur randomly. The customers arrive at the facility and wait in the queue (or buffer) if the server is not available. If there are many customers in the queue, they may suffer long delays which cause poor system performance. Thus, the arrival rate or the service rate may need to be controlled to reduce the delays. These systems may This research was supported in part DOE Grant DE-FG-96ER568. D.I. Choi was supported by the Korea Research Foundation. Current address of D.I. Choi: Department of Mathematics, Halla Institute of Technology, San 66 HeungUp-Li HeungUp-Myon, WonJu-Shi, KangWon-Do, -7, South Korea

be represented by queueing systems with queue-length-dependent arrival rates or service times. That is, if the queue length exceeds a threshold value, the arrival rate may be reduced (e.g. overload control), or the service rate may be increased (e.g. the cell discarding scheme []). Many schemes of traffic control in ATM (asynchronous transfer mode) networks have been analyzed using such threshold-based queueing systems ([],[5],[8]-[]). In this paper, we analyze a queueing system with queue-length-dependent service times. Customers arrive at the queue by a Poisson process, and there is only one server. The service times of customers depend on the queue length. Concretely, we specify a threshold value for the queue. If the queue length at service initiation of a customer is less than the threshold (respectively, greater than or equal to the threshold ), the service time of the customer follows a distribution with probability density function (respectively, ). We believe that our analysis can be extended to the case of multiple thresholds. Both infinite (M/G,G /) and finite capacity (M/G,G // ) queues are considered. The analysis of this queueing system was directly motivated by the cell-discarding scheme for voice packets in ATM networks (see [, 9, ]). In [], a system with deterministic service times was proposed, in which voice packets are divided into high and low priority ATM cells. The cells arrive as a concatenated pair (i.e. two cells per arrival) and the threshold and the capacity are measured in terms of cell pairs. The cell discarding occurs at the output of the queue and immediately prior to transmission, based upon the total number of cell pairs in the queue. If this number is less then then the next pair of cells is transmitted (no cells are discarded). However, if the queue length is greater than or equal to then only the high priority cell is transmitted. Thus, the low priority cell is discarded. The analysis in [] is based on numerically solving an embedded chain formulation of the problem. The system is studied only at service time completions. Our results, when specialized to G = D (deterministic service), are directly applicable to this model. We analyze this queueing system using the supplementary variable method. We first consider the case of an infinite capacity queue, and obtain an explicit formula for the steady-state queue length distribution. When the service times have different exponential distributions, the queue length distribution has a simple, closed form. We also compute asymptotic approximations to the queue length distribution for various choices of the system parameters. Next, we examine the finite capacity queue, and again obtain explicit expressions for and in particular the probability that the queue is full and the probability that it is empty. Also, we investigate asymptotic approximations for the queue length distribution for large values of the threshold and the queue capacity. We show that this queueing system has very different tail behavior (and hence loss probability) than other type models, and that the service tails can sometime determine the tail of the queue length. There has been some previous analytical work on queueing systems with queue-length-

dependent service times [, 3, 4, 6, 7]. C. M. Harris [6] considered the queue with queue-length-dependent service times. In particular, if there are customers in the queue, the service time of the customer starting service has a general distribution depending on. By using the embedded Markov chain method, C. M. Harris [6] derived the probability generating function for the queue length at the departure epochs. However, the obtained probability generating function contains infinitely many unknown constants. A closed form was obtained only for some special service times of two types. Fakinos [4] analyzed the queue in which the service discipline is last-come first-served and the service time depends on the queue length at the arrival epoch of each customer. Abolnikov, Dshalalow and Dukhovny [] considered queues with bulk arrivals (i.e. compound Poisson input) and state-dependent arrivals and service. Assuming that the statedependence applies only when the queue length is below a critical level, the authors characterize the generating function (for both transient and steady-state cases) of the queue length probabilities in terms of the roots of a certain equation. Ivnitskiy [7] also considers a model with bulk, statedependent arrivals and state-dependent service. Using the supplementary variable method, he obtains a recursion relation for the Laplace transforms of the transient queue length probabilities. For a very good recent survey of work on state-dependent queues, we refer the reader to Dshalalow [3]. The model here is a special case of that studied in [7]. However, we are able to give more explicit analytical expressions, from which we can easily obtain asymptotic expansions for tail probabilities and loss rates. These clearly show the qualitative dependence of these performance measures on the arrival rate and the service distribution(s). In particular, we show that the tail behavior of the model with threshold is much different than the tail behavior of the standard M/G/ and M/G//K models.. Statement of the Problem We let be the queue length at time, including the customer in service, and let be the remaining service time of the customer currently in service. Customers arrive according to a Poisson process with arrival rate and are served on a first-come first-serve basis. There is a single server and a queue with finite capacity. An arrival that would cause to exceed is lost, without effecting future arrivals. The service time of each customer depends on the queue length at the time that customer s service begins. If the queue length at service initiation is less than, the service time of that customer is, while if the queue length is greater than or equal to, the service time is. The service times and have density functions and with means and, respectively. We define and. The process is Markov and is referred to as a supplementary variable. 3

The stationary probabilities are denoted by lim (.) lim (.) For finite capacity systems ( ), these limits clearly exist. For infinite capacity systems ( ), we assume the stability condition The balance equations for (.)-(.) are (.3) (.4) (.5) (.6) (.7) (.8) (.9) The normalization condition is (.) For infinite capacity systems, we omit equation (.9) and let in (.8) and (.). An important local balance result can be obtained by integrating the balance equations with respect to from to, which leads to (.) In the following sections, we construct exact solutions to the infinite capacity model and then the finite capacity model. As we will show, the solution of the infinite capacity model can be used to construct the solution of the finite capacity model. Then we obtain simple formulas for the performance measures by constructing asymptotic approximations to the exact solutions. Infinite Capacity System ( ) We consider the infinite capacity model ( ) described by equations (.4)-(.8) with normalization condition (.). For this model, equation (.8) is valid for. 4

M/M,M / Queue: To illustrate the important characteristics of the solution, we first consider the case in which the service times, and, are exponential with probability density functions (.) The solution of (.4)-(.8) can be constructed in a straightfoward manner as follows. For, we assume a solution of (.4)-(.6) in the form which leads to the difference equation (.) (.3) where The solution of (.)-(.3) is (.4) (.5) so that (.6) We now compute by first finding the value of using (.6) with and the known functions and, which leads to (.7) We next solve (.7) for in terms of to obtain (.8) To compute, we substitute (.8) into the local balance result (.) with. This leads to which when used in (.8) gives (.9) For, we seek a solution of (.8) of the form (.) 5

which leads to the system of difference equations The solution of the above equations is where and are to be determined. By setting in (.), with and defined above, and equating the result to (.9), we find that Thus, for, we have The constant is determined by normalization using (.) and the marginals are summarzied below. Theorem M/M,M / Queue: Let and for and assume that the stability condition where is satisfied. The marginal probabilities are given by (.) An interesting aspect of the result is the tail behavior as has a different form depending on the parameters: (.) (.3). The tail probability if if (.4) 6

M/G,G / Queue: We now consider the system in which the service times and have general distributions. For, we solve (.4)-(.6) with by introducing the generating function (.5) into (.5)-(.6) to obtain (.6) where. The solution of (.6) is given by (.7) The unknown function is found by setting in (.7) to obtain ˆ ˆ (.8) where ˆ is the Laplace transform of. Combining the result (.8) for in (.7) and simplifying we obtain ˆ (.9) We invert to find an integral representation of the stationary probabilities in the form ˆ (.) The contour is a loop in the complex -plane which encircles the origin but excludes any other poles of the integrand. By setting and in (.), we find that (.4) is satisfied. Based on the above calculation, we see that (.) is in fact a solution of (.5)-(.6) for. Thus, we assume that for, the solution is of the form for some constant ˆ (.), which will be determined later. For, defining a different generating function (.) the equations (.7)-(.8) are transformed into (.3) 7

The solution of (.3) is As before, we determine by setting in (.4) to find that (.4) ˆ ˆ (.5) The numerator of (.5) vanishes as using the local balance result (.) with, i.e. (.6) We use the known solution for given by (.) to compute the integral in the numerator of (.5) as ˆ ˆ ˆ (.7) and using (.6) we find that for 3 ˆ (.8) Thus, is determined by (.5) using (.7) and (.8) up to the constant. We now use and to construct using (.4). After some algebra, we find that ˆ ˆ ˆ ˆ ˆ (.9) Thus, for, the stationary probabilities can be found by inverting (.) using (.3) The quantities of interest are the marginal probabilities the dependence on. We first compute which are obtained by removing 8

The following identity is useful in calculating the marginal probabilities ˆ We find that by integrating and simplifying that ˆ ˆ ˆ on (.3) In addition, we assume that 3 which is needed for the above simplification and to avoid a degenerate problem. We invert (.3) to obtain the marginal probabilities for, ˆ ˆ ˆ (.3) The contours for the double complex integral are such that on the -contour and both are small loops about the origin. The marginals for are obtained by integrating (.) with respect to which leads to ˆ ˆ (.33) and, using (.4), we find that ˆ ˆ (.34) Again the contours are small loops inside the unit circle. To complete the solution, we find the constant using the normalization condition (.). First, we compute ˆ ˆ ˆ where on the contour we choose. The only poles inside are at at and if. Since ˆ and ˆ, we find that ˆ (.35) where on the contour we have and. In, the function ˆ is assumed non-zero. 9

For, we must compute. We simplify (.33) as follows ˆ ˆ ˆ which leads to Combining the two sums, we find that ˆ (.36) ˆ ˆ (.37) We simplify (.37) by noting that if then ˆ ˆ so that ˆ (.38) By shifting the integration contour, we can remove the restriction that, and obtain the alternate form for in (.43). The final results for the marginals are summarized below. Theorem (M/G,G /) Let for be the density functions for the general service times with moments defined by (.39) and let and. We assume the stability condition is satisfied and that 3. The marginal probabilities are given by ˆ ˆ (.4) and for, ˆ ˆ ˆ (.4)

The probability that the system is empty is (.4) where ˆ All the integration contours are small loops about the origin. (.43) To illustrate the formulas in Theorem, we consider the case of exponential service times in which the density functions are defined in (.). The Laplace transforms of the density functions are defined by ˆ (.44) so that for, (.4) reduces to (.45) For, we must compute the double contour integral. For the exponential case, the formula (.4) becomes (.46) We compute the integral first by re-writting the above integral as (.47) The poles of the integral are located at so that this integral equals (Residue at ), which gives (.48) The remaining integral can be evaluated by observing that the integrand has poles at,, and. Since the only pole inside of is at, we have the equality: (Residue at ) = (Residue at ) + (Residue at ) so that (.48), after some simplification, reduces to (.49) The above results agree with (.) and (.) in Theorem. Also, the formula for in Theorem reduces to (.3).

Asymptotic Approximations: Simple formulas can be obtained by asymptotically expanding the exact integral representations for the stationary probabilities given in Theorem. An important limit is and. The asymptotic expansion in this limit depends on the location of the zeros of ˆ, ˆ, and poles of ˆ. Specifically, we need to locate the singularities of the integrand which are closest to the origin. Let be the non-zero solution of (.5) We assume that ˆ are analytic for some with, which insures the existence of a unique solution to (.5). Clearly, and satisfies if. We let be the solution of (.5) which satisfies when and as. The Laplace transform ˆ may have singularities in the half-plane. We assume the singularity with the largest real part is at and that ˆ (.5) for some constants. For example, if is exponential then ˆ,, and in (.5). When! Erlang, then ˆ so that,, i.e. -stage so that, and in (.5). If the service time is deterministic, i.e., then no singularity exists. To derive the asymptotic formula for an iterated integral and approximate the, we again view the double contour integral as integration first. The pole of to the origin is at where is the root of (.5). Thus, if, ˆ that is closest ˆ ˆ (.53) We define so that and (.54) ˆ ˆ (.55)

z-plane +B/# +C/# " Figure : A sketch of the -plane indicating the location of the poles of the integrand. +C/# z-plane! Figure : The contour!. If, there is no further simplification. However, if, then we can replace the integral in (.55) by an asymptotic approximation. The integrand is analytic at but has a pole at and (possibly) an algebraic singularity at, (see Figure ). The asymptotic approximation depends on the relative sizes of and. If, then the integral in (.55) can be approximated as [Residue at so that as ˆ (.56) where When, the singularity at of ˆ, (cf. (.5)), determines the dominant asymptotic behavior. The integral in (.55) can be approximated as! where the contour! is drawn in Figure. This leads to the following approximation for (.55) as 3

w-plane!$ Figure 3: The contour!. : ( ) ˆ! (.57) The integral can be further simplified by letting to obtain! where! is shown in Figure 3. Using the substitution can be expressed in terms of the Gamma function, so that for,!! ˆ! in the integral, we find that it We note that if ˆ has a simple pole at then and the algebraic factor disappears. that We derive an asymptotic formula for, and by using (.4) and noting ˆ ˆ 4 ˆ

We now compute asymptotic approximations to for. The constant is defined in Theorem as ˆ (.58) The poles of the integrand closest to are at and. We approximate when as (.59) Now if then and When, we find that so that (.6) (.6) The final case is when. We set and find that (.5) gives so that as. In addition, we find that Using the above, we find that if then exp (.6) The results are summarized below. Theorem 3 Asymptotic approximations for M/G,G / queue: For, and,, and defined by (.5)-(.5), the stationary probabilities have the following asymptotic approximations: and : ˆ (.63) 5

where and :! ˆ (.64) For and, and exp (.65) For and fixed (just above the threshold), the asymptotic result is given by (.55) with given above. We specialize the above formulas to the exponential service case with. For this case, we find that,,, and. We explicitly solve (.5) and (.5) to obtain for In addition, from (.5), we find that Using the above results, we obtain so that for, (.64) simplifies to which agrees with the result in (.4). We note that is equivalent to. A similar reduction occurs for the case when. 6

Finally, we examine the case when, more precisely. For, and, we have the approximation (.65). We let and note that the integral in (.55) now has singularities (at and ) that are close to each other. We consider the integral Integral We make the change of variables ˆ ˆ and use the approximations ˆ ˆ ˆ to obtain Integral! Here the contour! is shown in Figure 3 and it encircles both of the singularities of the integrand. The final approximation to (.55) is, for, (.66) where! If (i.e. ) then is explicitly evaluated as! 3 Finite Capacity System ( ) When the queue length has finite capacity, the stationary probabilities are solutions of the system (.4)-(.). The result for the infinite capacity system (.) is still valid for and the result (.3) is valid for except that must be recomputed taking into account that now. In addition, the probability must be computed be solving (.9), i.e. (3.) 7

We set in (.3) and use the result in (3.) to obtain ˆ (3.) ˆ ˆ ˆ Here we have used the fact that (3.3) The main quantities of interest are again the marginal probabilities. The marginals for, and are given by (.4) and (.4) in Theorem, respectively. Again, the constant must be re-calculated. The marginal probability must be computed using We use the identities and to obtain ˆ ˆ (3.4) ˆ ˆ ˆ ˆ ˆ ˆ This result can be further simplified by noting that 3 ˆ ˆ ˆ 8

and ˆ ˆ The final result is ˆ (3.5) ˆ ˆ ˆ The normalization constant is determined by (.), which we write as Substituting (.4), (.4) (with replaced by ), and (3.5) into (3.6), we find after simplifying that ˆ The results are summarized below. ˆ Theorem 4 (M/G,G //K) Let for be the density functions for the general service times as defined in Theorem and let ˆ 3. The marginal probabilities are given by ˆ (3.6) (3.7) (3.8) ˆ ˆ (3.9) and for, K-, ˆ ˆ ˆ (3.) The probability that the system is full is ˆ (3.) ˆ ˆ ˆ 9

The normalization constant, i.e. the probability that the system is empty, is ˆ ˆ ˆ ˆ (3.) On all the contours it is assumed that the origin is the only singularity within the contour. Asymptotic Approximations: We compute asymptotic approximations for the finite capacity system as and for various values of and. The asymptotic expansions of, except for and, are the same as in the infinite-capacity systems and are given in Theorem 3 for the two cases. Next, we expand given by (3.). The asymptotic approximation for the integrals depends on the location of the singularities of the integrand that are closest to the origin and follows closely the results for the infinite capacity system in the previous section (cf. the derviation of (.59)). For the single integral, these poles are at and, where if. The dominant poles of the double integral are located at, and where and if. Thus, for,, and, we find that uniformily in and.! (3.3) We can simplify (3.3) for different values of the parameters. For example, since, the last term in (3.3) is always negligible and may be dropped. The simplifications of (3.3) are summarized below.. and. and (3.4) (3.5) 3. and exp (3.6)

4. and (3.7) 5. and (heavy traffic limit) (3.8) 6. and (3.9) 7. and (3.) 8. and exp (3.) 9. and (a) : (b) : (c) : Note that in the last case has a bimodal behavior as, with peaks near and. The asymptotic expansion of as is computed by using (3.5) for and the expansions of the integrals which were derived for the infinite capacity case in Section. The results are summarized below. Theorem 5 Asymptotic expansions for the M/G,G //K queue: For and,, and defined by (.5)-(.5), the stationary probabilities have the asymptotic expansion (.65) for and (.63)- (.64) for. The asymptotic expansion for is! (3.) where the asymptotic expansions of are given by items -9 above for various values of and.

4 Discussion and Numerical Results We now demonstrate the usefulness of our results for a finite capacity system, namely the M/D,D // queue. For this model, the density functions of the service times are (4.) This model corresponds to cell-discarding model analyzed in [] if. The exact solution is given in Theorem 4 with (4.) in place of the general density functions. We illustrate how to numerically compute the exact solution. This calculation is only feasible for moderate and. For large values of and, we constructed the asymptotic approximations in Theorem 5. We demonstrate the accuracy of our asymptotic results by comparison with the exact solution. As we will see below, our asymptotic results are quite useful for moderate values of,, and and are extremely accurate when,, and are large. To evaluate the exact solution for in Theorem 4, we must compute the complex integrals in (3.9)-(3.). The simplest approach is to evaluate the integrals by using the method of residues. The residue at in (3.9) is difficult to compute if is large. This is a key motivation for the development of asymptotic expansions. For moderate values of, we compute the residues using the symbolic computation program Maple. The double complex integrals in (3.)-(3.) can be evaluated by re-writting the term (which couples the two integrals) as Using this result in the integrand in (3.) allows the double integral to be separated into where can be identified from (3.). The sum truncates at since for the integral vanishes. Finally, we must evaluate both integrals by computing the residues at and for each value of. Again the calculation is only feasible for and moderate in size. As before, we use Maple to perform the calculation. The calculation of the asymptotic approximation requires computing the constants and by solving (.5) and (.5), respectively. For deterministic service times, there is no singularity so that. Given the constants and, we evaluate the formulas (.65) and (.63) for if and (3.) for. The constant is given by (3.4)-(3.), depending on the values of and.

In Table, we consider a queue with the threshold 5 and capacity. The arrival rate is and the service times are and 4 so that and 4. Thus, when the queue length exceeds the threshold, the service time of the jobs entering service is half of the original service time. For these values of, the asymptotic value of is given by (3.4). The solutions of (.5) and (.5) are 5 and 9 346, respectively. Table Table 5 4 exact asympt. rel. err..57.5.44.3483.4783.4556.77.346.969 3.3784.383.5 4.9.9.56 5.43e-.943e-.883 6.453e-3.987e-3.8 7.56957e-4.8788e-4.54 8.687e-5.84885e-5.4434 9.749e-6.84e-6.555.6334e-7.6583e-7.3933 5 5 4 exact asympt. rel. err..57.5.44.3483.4783.4556.77.346.969 3.3784.383.5 4.9.9.56 5.43e-.943e-.883 6.453e-3.987e-3.8 7.56957e-4.8788e-4.54 8.687e-5.848857e-5.4434 9.749e-6.84e-6.555.7688e-7.7993e-7.45.75498e-8.76636e-8.57.7373e-9.74688e-9.4686 3.753e-.7587e-.69 4.6979e-.6988e-.9 5.5557e-.5558e-.48 As we see from Table, the asymptotic expansion is quite accurate for, 4 and 8. It is not accurate for and 5 8. This is consistent with our results in Theorem 5 since the asymptotic result (.65) is valid for and (.63) is valid for. It is remarkable that our results are this accurate since and are only moderate in size. We cannot expect the asymptotic solution to be accurate for since it assumes that. We have chosen values for and that are quite small. In reality, we would expect them to be of the order (see []). For such large values of and, calculation of the exact solution would prove difficult while the evaluation of the asymptotic solution is straightforward. In Table, we consider the same queue as in Table but now with 5. The values for and are the same as for Table since these constants are independent of and. We see that the exact and the asymptotic results are numerically close to those in the previous example for, as expected. For, the relative error starts at about 4% and rapidly decreases to 3

under %. The example in Table 3 is a queue with threshold 8 and capacity 5. The arrival rate is and the service times are and so that and. In this example, so that in the absence of the threshold, the queue length distribution would be peaked near the capacity. Now 7968 and 5. The asymptotic results are quite accurate when. However, when the results are not accurate. Table 4 8 5 exact asympt. rel. err..89e-5.89e-5.3.876e-4.869e-4.3397 Table 3.9756e-4.9986e-4.57 8 5 3.453.453.4 exact asympt. rel. err. 4.8.8..835e-5.89e-5.44 5.966.966.3.884e-4.869e-4.335 6.53968.53968.3.9796e-4.9986e-4.6 7.6565.6564.3 3.4586.457.33 8.9673 4.4998 4.4396 4.9.8.44 9.98693.78988 5.436998 5.97.965.44.437.36487.499576 6.5399.53967.44.45645.3644.7637 7.657.656.44.759.954.67786 8.98 4.499 4.397 3.65.8399.36569 9.9878.789 5.434 4..39.9578.48.3648.498 5.68.68.9.45665.364.696 6.85.94.49898.7598.95.67654 7.53894e-4.5554e-4.3365 3.656.83988.3658 8.554e-4.57e-4.98 4.4.398.9455 9.4455e-5.44694e-5.46 5.4394.47573.86589.7e-5.73e-5.65.3696e-6.368e-6.6.38e-6.3e-6.5 3.9348e-7.935e-7.64 4.83549e-8.8355e-8.7 5.664e-8.664e-8. In Table 4 we retain 8 and increase to 5. Now the asymptotic results are accurate to within 5% for 6 5. In Tables 3 and 4, the results are not accurate for 8 4. 4

Apparently, values. 6 is too small a value for (.63) to be useful, for these particular parameter In Table 5, we consider the same case as in Table 3 except that now 3 and hence 3. Now we have 7968 and 44. The results are to within 7% if 4 (i.e. 5). In Table 6 we increase to 5. The error is at most 7% for all 4 (i.e. 5). Table 5 8 5 3 exact asympt. rel. err..39e-5.9e-5.5.367e-4.47e-4.865.6973e-4.68989e-4.65 3.345.33953.54 4.6796.67.54 5.8667.84.54 6.4685.4475.54 7.3.99.54 8.5976.85983.3 9.33.43.8976.3486.874.3869.7473.8739.675.3895.469.68 3.8535.8978.39 4.883.8856.54 5.5867.579.95 Our results should be very accurate for Table 6 8 5 3 exact asympt. rel. err..9e-5.9e-5.5.3557e-4.47e-4.3397.6887e-4.68989e-4.55 3.34.34.3 4.67.67.4 5.84.84.5 6.4476.4476.5 7.994.993.5 8.5849.859833.3755 9.49.435.97433.3469.8747.3949.74339.874.73549.379.469.736 3.844.8978.93 4.8758.885.655 5.44.49.355 6.94.96.78 7.897.898.97 8.49.49.73 9.95.95.4.94e-4.94e-4..45e-4.45e-4 -.983e-4.983e-4-3.9457e-5.9457e-5-4.433e-5.433e-5-5.564e-5.564e-5-5, but it then becomes difficult to evaluate the exact solution. These numerical comparisons show that the asymptotic approximations are quite 5

robust, and are accurate even for relatively small values of and. Tables 4 and 6 show that when 5, the two results agree to five decimal places. Loss rates can thus be calculated to a very high precision using our asymptotic formulas. References [] L. ABOLNIKOV, J.H. DSHALALOW AND A. DUKHOVNY, On stochastic processes in a multilevel control bulk queueing system, Stoch. Anal. and Appl., 99, 55-79. [] B. D. CHOI AND D. I. CHOI, Queueing system with queue length dependent service times and its application to cell discarding scheme in ATM networks IEE Proc.-Commun. 43, Feb. 996, 5-. [3] J.H. DSHALALOW, Queueing systems with state-dependent parameters, in: Advances in Queueing (ed. by J.H. Dshalalow), 6-6, CRC Press, Boca Raton, FL, 995. [4] D. FAKINOS, The G/G/(LCFS/P) queue with service depending on queue size, Eur. J. Oper. Res. 59, 99, 33-37. [5] W. B. GONG, A. YAN AND C. G. CASSANDRAS, The M/G/ queue with queue-length dependent arrival rate, Stochastic Models 8, 99, 733-74. [6] C.M. HARRIS, Some results for bulk-arrival queue with state-dependent service times, Manage. Sci. 6, 97, 33-36. [7] V.A. IVNITSKIY, A stationary regime of a queueing system with parameters dependent on the queue length and with nonordinary flow, Eng. Cyb.,974, 85-9. [8] S. Q. LI, Overload control in a finite message storage buffer, IEEE Trans. Commun. 37, 989, 33-338. [9] K. SRIRAM, Methodologies for bandwidth allocation, transmission scheduling and congestion avoidance in broadband ATM networks, Comput. Netw. ISDN Syst. 6, 993, 43-59. [] K. SRIRAM AND D. M. LUCANTONI, Traffic smoothing of bit dropping in a packet voice multiplexer, IEEE Trans. Commun. 37, 989, 73-7. [] K. SRIRAM, R. S. MCKINNEY AND M. H. SHERIF, Voice packetisation and compression in broadband ATM networks, IEEE J. Sel. Areas Commun. 9(3), 99, 94-34. 6

[] H. TAKAGI, Analysis of a finite capacity M/G/ queue with resume level, Performance Evaluation 5, 985, 97-3. 7