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

Table 2

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

0	0.000892	0.005507	0.000981	0.000989	0.000996	0.001000	0.001002	0.001002	0.001001	0.000998	0.000993	0.000988	0.016350
1	0.001815	0.005277	0.000937	0.000941	0.000943	0.000943	0.000942	0.000939	0.000934	0.000928	0.000920	0.001897	0.017417
2	0.002765	0.005056	0.000894	0.000895	0.000893	0.000891	0.000886	0.000879	0.000872	0.000862	0.000853	0.002731	0.018480
3	0.003740	0.004845	0.000853	0.000851	0.000846	0.000840	0.000833	0.000824	0.000813	0.000802	0.000790	0.003497	0.019537
4	0.004735	0.004642	0.000814	0.000809	0.000802	0.000793	0.000783	0.000772	0.000759	0.000746	0.000732	0.004197	0.020587
5		0.004448	0.000778	0.000770	0.000760	0.000749	0.000736	0.000723	0.000709	0.000694	0.000678	0.004837	0.015881
10		0.003593	0.000616	0.000598	0.000580	0.000561	0.000541	0.000522	0.000501	0.000482	0.000463	0.007252	0.015710
50		0.000650	0.000096	0.000080	0.000067	0.000055	0.000046	0.000038	0.000032	0.000026	0.000021	0.007175	0.008288
100		0.000077	0.000009	0.000006	0.000004	0.000003	0.000002	0.000001	0.000001	0.000000	0.000000	0.002311	0.002417
200		0.000001	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000151	0.000152
300		0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000009	0.000009
370		0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000001	0.000001
380		0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
Total	0.013947	0.131681	0.021593	0.020178	0.018923	0.017798	0.016781	0.015856	0.015010	0.014231	0.013511	0.700489	1.000000
, , and
, , and