An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service
Abstract
:1. Introduction
2. Literature Survey
3. Model Description
Notations
- = at time t; the main server is in regular service, re-service and vacation, and at time t, the stand-by server is in service and idle, respectively.
- , if the server is on the j-th vacation.
- = number of customers in service station at time t.
- = number of customers in the queue at time t.
4. Queue Size Distribution
5. Probability Generating Function of the Queue Size
5.1. The PGF of the Queue Size at an Arbitrary Time Epoch
5.2. Steady State Condition
5.3. Computational Aspects
5.4. Result 1
5.5. Result 2
5.6. Particular Case
5.7. PGF of the Queue Size in Various Epochs
5.7.1. PGF of the Queue Size in the Main Server’s Service Completion Epoch
5.7.2. PGF of the Queue Size in the Vacation Completion Epoch
5.7.3. PGF of the Queue Size in the Main Server’s Re-Service Completion Epoch
5.7.4. PGF of the Queue Size in the Stand-by Server’s Service Completion Epoch
6. Some Performance Measures
6.1. The Main Server’s Expected Length of Idle Period
6.2. Expected Queue Length
6.3. Expected Waiting Time
7. Numerical Example
- The batch size distribution of the arrival is geometric with mean 2.
- Take a = 5 and b = 8, and the service time distribution is Erlang-2 (both servers).
- The vacation and re-service time of the main server follow an exponential distribution with parameter , respectively.
- Let be the service rate for the main server.
- Let be the service rate for the stand-by server.
- As arrival rate increases, the expected queue size and expected waiting time are also increase.
- When the main server’s and stand-by server’s service rate increases, the expected queue size and expected waiting time decrease.
- When the main server’s vacation rate increases, the expected queue size increases.
8. Conclusions
Author Contributions
Conflicts of Interest
Appendix A
Appendix B
References
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5.00 | 0.131407 | 8.657374 | 0.865737 |
5.25 | 0.137978 | 9.539724 | 0.908545 |
5.50 | 0.144548 | 10.375816 | 0.943256 |
5.75 | 0.151119 | 11.149076 | 0.969485 |
6.00 | 0.157689 | 11.843026 | 0.986919 |
6.25 | 0.164259 | 12.441064 | 0.995285 |
6.50 | 0.170830 | 12.927026 | 0.994387 |
6.75 | 0.177400 | 13.285663 | 0.984123 |
7.00 | 0.183970 | 13.502456 | 0.964461 |
7.25 | 0.190541 | 13.563622 | 0.935422 |
7.50 | 0.197111 | 13.456834 | 0.897122 |
5.25 | 0.257410 | 26.824821 | 2.682482 |
5.50 | 0.248885 | 26.082729 | 2.608273 |
5.75 | 0.240905 | 25.388555 | 2.538855 |
6.00 | 0.233420 | 24.735574 | 2.473557 |
6.25 | 0.226385 | 24.117976 | 2.411798 |
6.50 | 0.219761 | 23.531054 | 2.353105 |
6.75 | 0.213513 | 22.971202 | 2.297120 |
7.00 | 0.207610 | 22.435611 | 2.243561 |
7.25 | 0.202024 | 21.921511 | 2.192151 |
7.50 | 0.196731 | 21.427009 | 2.142701 |
7.75 | 0.191707 | 20.950354 | 2.095035 |
8.00 | 0.186934 | 20.490051 | 2.049005 |
4.0 | 0.260103 | 65.007246 | 4.062953 |
4.5 | 0.254645 | 59.415276 | 3.713455 |
5.0 | 0.249401 | 53.808260 | 3.363016 |
5.5 | 0.244361 | 48.320593 | 3.020037 |
6.0 | 0.239515 | 43.028238 | 2.689265 |
6.5 | 0.234853 | 37.970425 | 2.373152 |
7.0 | 0.230366 | 33.162455 | 2.072653 |
7.5 | 0.226045 | 28.605406 | 1.787838 |
8.0 | 0.22188 | 24.292435 | 1.518277 |
8.5 | 0.217865 | 20.212075 | 1.263255 |
9.0 | 0.213991 | 16.351005 | 1.021938 |
Erlang | Exponential | |
---|---|---|
5.00 | 8.657374 | 8.279153 |
5.25 | 8.808004 | 8.448114 |
5.50 | 8.950939 | 8.607757 |
5.75 | 9.086640 | 8.758748 |
6.00 | 9.215535 | 8.901685 |
6.25 | 9.338068 | 9.037158 |
6.50 | 9.454625 | 9.165674 |
6.75 | 9.565595 | 9.287730 |
7.00 | 9.671330 | 9.403767 |
7.25 | 9.772165 | 9.514200 |
7.50 | 9.868410 | 9.619407 |
7.75 | 9.960351 | 9.719734 |
8.00 | 10.048247 | 9.815494 |
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Ayyappan, G.; Karpagam, S. An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service. Mathematics 2018, 6, 101. https://doi.org/10.3390/math6060101
Ayyappan G, Karpagam S. An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service. Mathematics. 2018; 6(6):101. https://doi.org/10.3390/math6060101
Chicago/Turabian StyleAyyappan, G., and S. Karpagam. 2018. "An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service" Mathematics 6, no. 6: 101. https://doi.org/10.3390/math6060101