Queue Length and Server Content Distribution in an Infinite-Buffer Batch-Service Queue with Batch-Size-Dependent Service

Table 6

Joint distribution of queue and server content at arbitrary epoch () of an queue with .

0	0.000929	0.071400	0.021036	0.018260	0.014403	0.010397	0.006902	0.004231	0.147558
1	0.005022	0.071016	0.020723	0.017708	0.013675	0.009621	0.006204	0.006122	0.150090
2	0.014470	0.069633	0.019782	0.016289	0.012037	0.008069	0.004948	0.006222	0.151450
3	0.029819	0.066314	0.017900	0.013857	0.009580	0.006000	0.003440	0.005221	0.152130
4	0.049663	0.060341	0.015077	0.010730	0.006815	0.003931	0.002083	0.003812	0.152452
5		0.051739	0.011690	0.007514	0.004327	0.002276	0.001106	0.002489	0.081141
6		0.041417	0.008303	0.004757	0.002461	0.001172	0.000520	0.001482	0.060112
7		0.030800	0.005399	0.002731	0.001261	0.000541	0.000218	0.000816	0.041766
8		0.021245	0.003221	0.001429	0.000586	0.000226	0.000083	0.000421	0.027211
9		0.013601	0.001770	0.000685	0.000249	0.000086	0.000028	0.000205	0.016624
10		0.008097	0.000899	0.000302	0.000097	0.000030	0.000009	0.000094	0.009528
12		0.002333	0.000186	0.000047	0.000012	0.000003	0.000000	0.000018	0.002599
14		0.000520	0.000029	0.000005	0.000001	0.000000	0.000000	0.000003	0.000558
16		0.000092	0.000004	0.000000	0.000000	0.000000	0.000000	0.000000	0.000096
18		0.000013	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000013
20		0.000001	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000001
23		0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
Total	0.099903	0.514462	0.126533	0.094459	0.065544	0.042364	0.025546	0.031189	1.000000
, , and
, , and