# Departure Process in Finite-Buffer Queue with Batch Arrivals (Queueing Theory) (Analytical and Stochastic Modeling) Part 2

## Numerical Results

In this section we present sample numerical results for conditional mean of departures before fixed t i.e. for the expression Enh(t), where the symbol En stands for the mean on condition £(0) = n. In computations we use firstly the following obvious identity:

and next the algorithm of approximate numerical Laplace transform inversion introduced in . The base of the algorithm is the Bromwich inversion integral that finds value of the function g at fixed t > 0 from its transform g as

where is located on the right to all singularities of g. Using trapezoidal method with step h to estimate the integral in (47) we obtain the approximation gh(t) of g(t) in the form

Setting series representation:

where

and

Let us take into consideration the following Euler formula for approximate summation of alternating series:

where parameter m is usually of order of dozen and parameter n – of order of several dozen (). Applying (53) in (48) we finally obtain

where typical values of parameters are following (see [l]): m = 38,n = ll,A = l9, L = l.

A precise evaluation of the estimation error in (54) is not possible, first of all, due to the lack of such an estimation for the Euler summation formula (53). The practice has shown that a good evaluation of the error one can get by executing the calculation twice, and changing by l one of the parameters. Then the difference between the results, e.g. for A = l9 and A = 20 provides a good evaluation of the error estimation. More details can be found in [l] and .

Let us consider the system of size N = 4 in that individual service times have Erlang distributions with parameters n = 2 (shape) and ^ = l (rate) i.e.

thus the mean service time is 2. Let us take into consideration two different simple batch size distributions: p2 = l (each arriving batch consists of 2 packets) and pi = p2 = The system with the second possibility we denote by M1,2/G/l/N. Below we present a numerical comparison of values E0h(t) and ENh(t), computed for t = 20, 50, l00, 200, 500 and l000 for these two systems. Possibilities of underloaded (p < l) and overloaded system (p > l) are presented separately. Each time values of the arrival rate parameter A are selected in such a way to fix the same value of p for both systems.

In the table below (Table l) we present results obtained for A1 = 0.l5 (for M2/G/l/N queue) and A12 = 0.20 (for M1’2/G/l/N queue) that gives q = 0.6 < l in both systems.

Table 1. Conditional means of departures before t for p = 0.6 (underloaded systems)   t    20 4.299 7.133 4.479 7.334 50 12.204 15.059 12.632 15.523 100 25.384 28.240 26.231 29.121 200 51.745 54.600 53.427 56.318 500 130.827 133.682 135.016 137.907 1000 262.630 265.486 270.998 273.889

Table 2. Conditional means of departures before t for p = 2.4 (overloaded systems)   t    20 8.559 9.495 8.798 9.601 50 22.997 23.933 23.470 24.273 100 47.061 47.996 47.922 48.726 200 95.187 96.123 96.828 97.631 500 239.568 240.503 243.545 244.348 1000 480.202 481.138 488.073 488.876

The next table present similar results for Ai = 0.6 and A1,2 = 0.8 (so for p = 2.4 > 1 in both systems).

Considering the above results one can observe that the average number of departures is greater for the M1,2/G/1/N system (for the same p). But the difference is the smaller, the greater is traffic load p. Similarly, of course, the mean of h(t) is greater in the case of the opening with maximum number of customers present than in the case of the system empty at t = 0. The difference decreases as t increases.