The Discrete-Time Bulk-Service Geo/Geo/1 Queue with Multiple Working Vacations

Table 1

Queue length distributions at random slots for 𝑎=1, 𝑝=0.3, 𝜇_𝑣=0.4, 𝜇_𝑏=0.6, and 𝜃=0.7.

𝑛	𝜋𝑛,0	𝜋𝑛,1	𝑛	𝜋𝑛,0	𝜋𝑛,1
0	0.5580	0	0	0.3906	0
1	0.0363	0.3128	1	0.0603	0.3539
2	0.0022	0.0679	2	0.0037	0.1501
3	1.39𝐸−04	0.0128	3	2.32𝐸−04	0.0299
4	8.64𝐸−06	0.0023	4	1.44𝐸−05	0.0055
5	5.35𝐸−07	4.17𝐸−04	5	8.90𝐸−07	9.92𝐸−04
6	3.32𝐸−08	7.43𝐸−05	6	5.51𝐸−08	1.77𝐸−04
7	2.05𝐸−09	1.32𝐸−05	7	3.42𝐸−09	3.16𝐸−05
8	1.27𝐸−10	2.35𝐸−06	8	2.12𝐸−10	5.62𝐸−06
9	7.89𝐸−12	4.19𝐸−07	9	1.31𝐸−11	1.00𝐸−06
10	4.89𝐸−13	7.45𝐸−08	10	8.13𝐸−13	1.78𝐸−07
11	3.03𝐸−14	1.32𝐸−08	11	5.04𝐸−14	3.16𝐸−08
12	1.88𝐸−15	2.36𝐸−09	12	3.12𝐸−15	5.62𝐸−09
13	1.16𝐸−16	4.19𝐸−10	13	1.93𝐸−16	1.00𝐸−09
14	7.21𝐸−18	7.45𝐸−11	14	1.20𝐸−17	1.78𝐸−10
15	4.47𝐸−19	1.33𝐸−11	15	7.43𝐸−19	3.16𝐸−11
Sum	1		Sum	1

𝐸(𝐿)=0.2741, 𝐸(𝑤_𝑞)=1.2783.