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Transient analysis of a queue with system disasters and customer impatience. (English) Zbl 1197.60087
Summary: A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time, any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter \(\xi \). If the system is not repaired before the customer’s timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions.

MSC:
60K25 Queueing theory (aspects of probability theory)
30B70 Continued fractions; complex-analytic aspects
90B22 Queues and service in operations research
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[1] Abate, J., Whitt, W.: Computing Laplace transforms for numerical inversion via continued fractions. INFORMS J. Comput. 11, 394–405 (1999) · Zbl 1039.65512 · doi:10.1287/ijoc.11.4.394
[2] Altman, E., Yechiali, U.: Analysis of customer’s impatience in queues with server vacations. Queueing Syst. 52, 261–279 (2006) · Zbl 1114.90015 · doi:10.1007/s11134-006-6134-x
[3] Bowman, K.O., Shenton, L.R.: Continued Fractions in Statistical Applications. Dekker, New York (1989) · Zbl 0702.62004
[4] Chakravarthy, S.R.: A disaster queue with Markovian arrivals and impatient customers. Appl. Math. Comput. 214, 48–59 (2009) · Zbl 1170.60330 · doi:10.1016/j.amc.2009.03.081
[5] Chen, A., Renshaw, E.: The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Probab. 34(1), 192–207 (1997) · Zbl 0876.60079 · doi:10.2307/3215186
[6] Gelenbe, E.: Product form networks with negative and positive customers. J. Appl. Probab. 28, 655–663 (1991) · Zbl 0741.60091
[7] Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 6th edn. Academic Press, New York (2000) · Zbl 0981.65001
[8] Jain, G., Sigman, K.: A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J. Appl. Probab. 33(4), 1191–1200 (1996) · Zbl 0867.60082 · doi:10.2307/3214996
[9] Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. Studies in Computational Mathematics, vol. 3. Elsevier, Amsterdam (1992) · Zbl 0782.40001
[10] Parthasarathy, P.R., Lenin, R.B.: Birth and Death Process (BDP) Models with Applications–Queueing, Communication Systems, Chemical Models, Biological Models: The State-of-the-Art with a Time-Dependent Perspective. American Series in Mathematical and Management Sciences, vol. 51. American Sciences Press, Columbus (2004) · Zbl 1065.60120
[11] Parthasarathy, P.R., Selvaraju, N.: Transient analysis of a queue where potential customers are discouraged by queue length. Math. Probl. Eng. 7, 433–454 (2001) · Zbl 1008.60098 · doi:10.1155/S1024123X01001727
[12] Parthasarathy, P.R., Sudhesh, R.: Exact transient solution of state-dependent birth-death processes. J. Appl. Math. Stoch. Anal. 2006, 1–16 (2006). Article ID 97073 · Zbl 1102.60073 · doi:10.1155/JAMSA/2006/97073
[13] Parthasarathy, P.R., Sudhesh, R.: Exact transient solution of a discrete time queue with state-dependent rates. Am. J. Math. Manag. Sci. 26, 253–276 (2006) · Zbl 1244.90068
[14] Parthasarathy, P.R., Sudhesh, R.: Time-dependent analysis of a single-server retrial queue with state-dependent rates. Oper. Res. Lett. 35, 601–611 (2007) · Zbl 1149.90042 · doi:10.1016/j.orl.2006.12.005
[15] Yechiali, U.: Queues with system disasters and impatient customers when system is down. Queueing Syst. 56, 195–202 (2007) · Zbl 1124.60076 · doi:10.1007/s11134-007-9031-z
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