Availability and profit analysis of a two-unit cold standby system for general distribution

Figures & data

Figure 1. State transition diagram.

Table 1. Effect of failure rate of new unit and other parameters for Weibull distribution on MTSF

MTSF
λ	Weibull distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	387.285	387.260	387.891	387.563	387.291	387.327
0.02	191.794	191.770	192.153	191.874	191.800	191.835
0.03	127.092	127.068	127.346	127.172	127.098	127.131
0.04	94.911	94.888	95.1054	94.920	94.917	94.950
0.05	75.686	75.664	75.841	75.721	75.692	75.725
0.06	62.918	62.895	63.044	62.929	62.923	62.956
0.07	53.827	53.805	53.932	53.933	53.833	53.864
0.08	47.029	47.007	47.117	47.067	47.034	47.066
0.09	41.755	41.734	41.830	41.872	41.761	41.792
0.1	37.546	37.525	37.610	37.726	37.552	37.582

Table 2. Effect of failure rate of new unit and other parameters for exponential distribution on MTSF

MTSF
λ	Exponential distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	200.621	200.615	200.657	200.631	200.622	200.637
0.02	100.617	100.611	100.653	100.627	100.618	100.633
0.03	67.280	67.274	67.315	67.289	67.281	67.295
0.04	50.609	50.604	50.644	50.619	50.611	50.625
0.05	40.605	40.600	40.640	40.615	40.607	40.621
0.06	33.935	33.930	33.969	33.945	33.937	33.951
0.07	29.170	29.164	29.203	29.179	29.171	29.185
0.08	25.595	25.589	25.628	25.604	25.596	25.610
0.09	22.813	22.808	22.846	22.823	22.815	22.829
0.1	20.588	20.583	20.620	20.597	20.589	20.603

Table 3. Effect of failure rate of new unit and other parameters for Rayleigh distribution on MTSF

MTSF
λ	Rayleigh distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	178.353	178.353	178.426	178.361	178.355	178.355
0.02	89.727	89.726	89.799	89.735	89.729	89.729
0.03	60.182	60.182	60.254	60.190	60.185	60.185
0.04	45.408	45.408	45.480	45.416	45.411	45.410
0.05	36.542	36.542	36.613	36.550	36.544	36.544
0.06	30.630	30.630	30.701	30.638	30.633	30.632
0.07	26.406	26.406	26.476	26.414	26.409	26.408
0.08	23.237	23.237	23.307	23.245	23.239	23.239
0.09	20.771	20.771	20.840	20.779	20.774	20.774
0.1	18.798	18.797	18.866	18.806	18.800	18.800

Table 4. Effect of failure rate of new unit and other parameters for Weibull distribution on availability

Availability
λ	Weibull distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	0.995043	0.994830	0.995379	0.995187	0.996051	0.995638
0.02	0.989780	0.989341	0.990487	0.990063	0.991877	0.990986
0.03	0.984403	0.983731	0.985493	0.984818	0.987609	0.986213
0.04	0.978982	0.978076	0.980458	0.979523	0.983296	0.981382
0.05	0.973559	0.972416	0.975418	0.974218	0.978968	0.976528
0.06	0.968158	0.966781	0.970397	0.968930	0.974642	0.971676
0.07	0.962799	0.961188	0.965411	0.963677	0.970333	0.966844
0.08	0.957495	0.955652	0.960471	0.958474	0.966051	0.962042
0.09	0.952256	0.950184	0.955587	0.953330	0.961802	0.957282
0.1	0.947089	0.944790	0.950766	0.948253	0.957594	0.952571

Table 5. Effect of failure rate of new unit and other parameters for exponential distribution on availability

Availability
λ	Exponential distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	0.999293	0.999200	0.999374	0.999297	0.999364	0.999339
0.02	0.998587	0.998401	0.998751	0.998596	0.998732	0.998678
0.03	0.997883	0.997603	0.998129	0.997895	0.998102	0.998018
0.04	0.997180	0.996808	0.997510	0.997196	0.997475	0.997359
0.05	0.996480	0.996015	0.996893	0.996499	0.996852	0.996701
0.06	0.995781	0.995224	0.996278	0.995804	0.996231	0.996045
0.07	0.995085	0.994436	0.995667	0.995111	0.995615	0.995390
0.08	0.994392	0.993651	0.995058	0.994420	0.995001	0.994738
0.09	0.993702	0.992870	0.994453	0.993733	0.994391	0.994087
0.1	0.993015	0.992093	0.993851	0.993048	0.993786	0.993439

Table 6. Effect of failure rate of new unit and other parameters for Rayleigh distribution on availability

Availability
λ	Rayleigh distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	0.999975	0.999967	0.999981	0.999976	0.999976	0.999976
0.02	0.999952	0.999936	0.999963	0.999953	0.999953	0.999952
0.03	0.999928	0.999905	0.999946	0.999931	0.999930	0.999929
0.04	0.999906	0.999875	0.999929	0.999909	0.999908	0.999907
0.05	0.999883	0.999845	0.999912	0.999888	0.999886	0.999885
0.06	0.999861	0.999816	0.999895	0.999866	0.999865	0.999863
0.07	0.999840	0.999787	0.999879	0.999845	0.999844	0.999842
0.08	0.999818	0.999759	0.999863	0.999825	0.999823	0.999821
0.09	0.999797	0.999731	0.999847	0.999804	0.999803	0.999800
0.1	0.999777	0.999704	0.999831	0.999784	0.999783	0.999780

Table 7. Effect of failure rate of new unit and other parameters for Weibull distribution on profit

Profit
λ	Weibull distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	4,970.234	4,969.169	4,972.255	4,970.870	4,975.649	4,973.779
0.02	4,938.825	4,936.626	4,943.051	4,940.060	4,950.045	4,945.991
0.03	4,906.839	4,903.479	4,913.327	4,908.637	4,923.946	4,917.576
0.04	4,874.667	4,870.133	4,883.426	4,876.992	4,897.630	4,888.882
0.05	4,842.534	4,836.823	4,853.546	4,845.351	4,871.261	4,860.111
0.06	4,810.585	4,803.700	4,823.818	4,813.861	4,844.946	4,831.394
0.07	4,778.923	4,770.869	4,794.332	4,782.624	4,818.762	4,802.826
0.08	4,747.619	4,738.407	4,765.153	4,751.716	4,792.766	4,774.477
0.09	4,716.729	4,706.369	4,736.330	4,721.191	4,767.000	4,746.400
0.1	4,686.291	4,674.796	4,707.900	4,691.090	4,741.500	4,718.634

Table 8. Effect of failure rate of new unit and other parameters for exponential distribution on profit

Profit
λ	Exponential distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	4,992.230	4,991.755	4,992.933	4,992.985	4,992.755	4,992.779
0.02	4,984.519	4,983.571	4,985.926	4,984.633	4,985.566	4,985.603
0.03	4,976.869	4,975.450	4,978.980	4,976.944	4,978.434	4,978.474
0.04	4,969.282	4,967.391	4,972.095	4,970.321	4,971.360	4,971.394
0.05	4,961.759	4,959.398	4,965.271	4,962.564	4,964.344	4,964.365
0.06	4,954.300	4,951.472	4,958.511	4,955.874	4,957.388	4,957.386
0.07	4,946.908	4,943.614	4,951.813	4,948.254	4,950.492	4,950.461
0.08	4,939.583	4,935.824	4,945.179	4,941.703	4,943.656	4,943.589
0.09	4,932.325	4,928.105	4,938.609	4,935.223	4,936.882	4,936.773
0.1	4,925.136	4,920.456	4,932.104	4,928.814	4,930.170	4,930.012

Table 9. Effect of failure rate of new unit and other parameters for Rayleigh distribution on profit

Profit
λ	Rayleigh distribution
	λ1 = 0.13	λ1 = 0.15	r = 1.8	r2 = 5	θ = 12	p = 0.4, q = 0.6
0.01	4,996.330	4,996.285	4,996.670	4,996.430	4,996.425	4,996.396
0.02	4,992.734	4,992.646	4,993.403	4,992.832	4,992.920	4,992.865
0.03	4,989.211	4,989.080	4,990.198	4,990.437	4,989.483	4,989.403
0.04	4,985.756	4,985.583	4,987.051	4,986.342	4,986.111	4,986.007
0.05	4,982.367	4,982.154	4,983.962	4,983.564	4,982.802	4,982.676
0.06	4,979.042	4,978.788	4,980.927	4,979.409	4,979.552	4,979.405
0.07	4,975.777	4,975.483	4,977.944	4,976.030	4,976.359	4,976.193
0.08	4,972.571	4,972.238	4,975.012	4,973.131	4,973.222	4,973.037
0.09	4,969.420	4,969.049	4,972.129	4,970.478	4,970.138	4,969.936
0.1	4,966.324	4,965.915	4,969.293	4,967.309	4,967.105	4,966.886