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

Table 4

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

0	0.000253	0.019330	0.004538	0.004620	0.004596	0.004487	0.004311	0.004089	0.003835	0.003563	0.053622
1	0.001016	0.019258	0.004511	0.004580	0.004540	0.004414	0.004222	0.003983	0.003715	0.006723	0.056962
2	0.002458	0.019111	0.004458	0.004502	0.004436	0.004282	0.004064	0.003802	0.003513	0.009404	0.060030
3	0.004637	0.018872	0.004373	0.004382	0.004279	0.004090	0.003840	0.003550	0.003241	0.011560	0.062824
4	0.007525	0.018529	0.004256	0.004221	0.004075	0.003846	0.003562	0.003247	0.002921	0.013182	0.065364
5	0.011027	0.018082	0.004108	0.004023	0.003830	0.003561	0.003247	0.002913	0.002576	0.014290	0.067657
6	0.015016	0.017535	0.003932	0.003795	0.003555	0.003251	0.002913	0.002567	0.002229	0.014930	0.069723
7		0.016896	0.003732	0.003543	0.003262	0.002929	0.002575	0.002226	0.001896	0.015162	0.052221
8		0.016180	0.003515	0.003277	0.002961	0.002606	0.002246	0.001902	0.001588	0.015054	0.049329
9		0.015400	0.003286	0.003005	0.002660	0.002294	0.001936	0.001605	0.001312	0.014674	0.046172
10		0.014572	0.003049	0.002732	0.002368	0.001998	0.001650	0.001338	0.001070	0.014088	0.042865
20		0.006445	0.001064	0.000748	0.000508	0.000336	0.000218	0.000139	0.000088	0.005824	0.015370
30		0.002052	0.000258	0.000138	0.000072	0.000036	0.000018	0.000009	0.000004	0.001642	0.004229
40		0.000537	0.000051	0.000020	0.000008	0.000003	0.000001	0.000000	0.000000	0.000397	0.001017
60		0.000026	0.000001	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000018	0.000045
75		0.000002	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000001	0.000003
85		0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
Total	0.041932	0.344036	0.069281	0.061779	0.054699	0.048127	0.042119	0.036691	0.031838	0.269497	1.000000
, , and
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