## 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 [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

whereis 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 ([3]). 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 [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.

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.