Performance analysis and optimization of a retrial queue with working vacations and starting failures

Figures & data

Figure 1. Basic model for an M/M/1/WV retrial queue with starting failures.

Figure 2. The state transition diagram for an M/M/1/WV retrial queue with starting failures.

Figure 3. Effect of $\lambda$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 1).

Figure 4. Effect of ${\mu }_v$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 2).

Figure 5. Effect of ${\mu }_b$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 3).

Figure 6. Effect of $\beta$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 4).

Figure 7. Effect of $\nu$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 5).

Figure 8. Effect of $\gamma$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 6).

Figure 9. Effect of $\delta$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 7).

Figure 10. Effect of $\theta$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$ (case 8).

Table 1. Optimal service rates $({\mu }_v^{\ast },{\mu }_b^{\ast })$ and the corresponding expected cost $F({\mu }_v^{\ast },{\mu }_b^{\ast })$ for different values of $\lambda$.

$\lambda$	0.5	1.0	1.5	2.0	2.5	3.0
${\mu }_v^{\ast }$	1.4932	2.1842	2.8395	3.4717	4.0449	4.3651
${\mu }_b^{\ast }$	1.5810	3.6002	6.7294	11.8441	21.2200	30.0000
$F({\mu }_v^{\ast },{\mu }_b^{\ast })$	189.796	290.395	432.258	661.191	1083.050	2301.318
Iterations	47	64	47	62	59	54
CPU time	43.091	100.785	95.806	161.140	203.165	341.029

Table 2. Optimal service rates $({\mu }_v^{\ast },{\mu }_b^{\ast })$ and the corresponding expected cost $F({\mu }_v^{\ast },{\mu }_b^{\ast })$ for different values of $\beta$.

$\beta$	0.5	0.7	0.9	1.1	1.3	1.5
${\mu }_v^{\ast }$	2.2472	2.1887	2.1820	2.1879	2.1966	2.2055
${\mu }_b^{\ast }$	6.8209	4.6189	3.8268	3.4315	3.1986	3.0465
$F({\mu }_v^{\ast },{\mu }_b^{\ast })$	448.016	335.273	299.848	283.586	274.546	268.895
Iterations	56	85	115	173	61	48
CPU time	124.829	138.64	195.984	232.822	68.648	2953.08

Table 3. Optimal service rates $({\mu }_v^{\ast },{\mu }_b^{\ast })$ and the corresponding expected cost $F({\mu }_v^{\ast },{\mu }_b^{\ast })$ for different values of $v$.

$v$	0.5	0.7	0.9	1.1	1.3	1.5
${\mu }_v^{\ast }$	1.2032	0.7135	0.2945	0	0	0
${\mu }_b^{\ast }$	3.9246	4.0103	4.0577	4.0846	4.1082	4.1260
$F({\mu }_v^{\ast },{\mu }_b^{\ast })$	274.079	265.080	257.431	250.880	245.968	242.385
Iterations	45	62	47	46	51	58
CPU time	65.869	82.761	68.052	65.692	71.943	80.614

Table 4. Optimal service rates $({\mu }_v^{\ast },{\mu }_b^{\ast })$ and the corresponding expected cost $F({\mu }_v^{\ast },{\mu }_b^{\ast })$ for different values of $\gamma$.

$\gamma$	0.5	1.0	1.5	2.0	2.5	3.0
${\mu }_v^{\ast }$	1.2349	1.8555	2.0781	2.1843	2.2437	2.2809
${\mu }_b^{\ast }$	5.1534	4.2113	3.8171	3.6002	3.4634	3.3695
$F({\mu }_v^{\ast },{\mu }_b^{\ast })$	340.251	309.048	297.049	290.395	286.118	283.128
Iterations	95	40	101	43	52	89
CPU time	135.811	50.677	136.717	59.182	70.484	118.308

Table 5. Optimal service rates $({\mu }_v^{\ast },{\mu }_b^{\ast })$ and the corresponding expected cost $F({\mu }_v^{\ast },{\mu }_b^{\ast })$ for different values of $\delta$.

$\delta$	0.3	0.4	0.5	0.6	0.7	0.8
${\mu }_v^{\ast }$	0.5793	0.8332	1.1059	1.3884	1.6637	1.9222
${\mu }_b^{\ast }$	3.8856	3.8804	3.8731	3.8562	3.8191	3.7439
$F({\mu }_v^{\ast },{\mu }_b^{\ast })$	370.612	361.847	351.046	338.241	323.665	307.700
Iterations	45	42	55	50	86	52
CPU time	86.399	73.611	94.646	78.236	125.566	74.321

Table 6. Optimal service rates $({\mu }_v^{\ast },{\mu }_b^{\ast })$ and the corresponding expected cost $F({\mu }_v^{\ast },{\mu }_b^{\ast })$ for different values of $\theta$.

$\theta$	0.55	0.6	0.7	0.8	0.9	1.0
${\mu }_v^{\ast }$	3.4578	2.8021	2.3038	2.1842	2.1680	2.1843
${\mu }_b^{\ast }$	10.0000	10.0000	5.2442	3.6002	2.8435	2.4279
$F({\mu }_v^{\ast },{\mu }_b^{\ast })$	1772.060	574.944	346.610	290.395	268.831	258.717
Iterations	44	97	65	72	59	135
CPU time	268.664	238.092	108.138	99.508	71.901	147.042

Figure 11. Expected cost $F({\mu }_v,{\mu }_b)$ for various values of ${\mu }_v$ and ${\mu }_b$.

Table 7. Optimal service rates $({\mu }_v^{\ast },{\mu }_b^{\ast })$ and the corresponding expected cost $F({\mu }_v^{\ast },{\mu }_b^{\ast })$ for different values of $\tau$.

$\tau$	0.5	0.6	0.7	0.8	0.9	1.0
${\mu }_v^{\ast }$	1.9544	2.1495	2.3090	2.4291	2.5209	2.5341
${\mu }_b^{\ast }$	15.3794	6.4964	4.3709	3.4129	2.8591	2.5341
$F({\mu }_v^{\ast },{\mu }_b^{\ast })$	733.521	391.981	316.170	282.770	263.354	250.272
Iterations	63	129	80	75	81	146
CPU time	189.996	202.83	81.788	64.783	57.212	53.376

Figure 12. Effect of $\tau$ on $L_S$, $P_V$, $P_D$, $P_I$, and $P_B$.