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:

tmp4A2-71_thumb[2]

and next the algorithm of approximate numerical Laplace transform inversion introduced in [1]. 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

tmp4A2-72_thumb[2]

wheretmp4A2-73_thumb[2]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


tmp4A2-75_thumb[2] 

Setting tmp4A2-76_thumb[2] series representation:

 

tmp4A2-77_thumb[2]

where

tmp4A2-78_thumb[2]

and

tmp4A2-79_thumb[2]

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

tmp4A2-80_thumb[2]

where parameter m is usually of order of dozen and parameter n – of order of several dozen ([3]). Applying (53) in (48) we finally obtain

tmp4A2-81_thumb[2]

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 [3].

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.

tmp4A2-82_thumb[2]

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)

tmp4A2-83 tmp4A2-84 tmp4A2-85

t

tmp4A2-86 tmp4A2-87 tmp4A2-88 tmp4A2-89

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)

tmp4A2-90 tmp4A2-91 tmp4A2-92

t

tmp4A2-93 tmp4A2-94 tmp4A2-95 tmp4A2-96

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.

Next post:

Previous post: