## Queues with slow servers and impatient customers.(English)Zbl 1177.90100

Summary: We study $$M/M/c$$ queues ($$c=1, 1<c<\infty$$ and $$c=\infty )$$ in a 2-phase (fast and slow) Markovian random environment, with impatient customers. The system resides in the fast phase (phase 1) an exponentially distributed random time with parameter $$\eta$$ and the arrival and service rates are $$\lambda$$ and $$\mu$$, respectively. The corresponding parameters for the slow phase (phase 0) are $$\gamma, \lambda _{0}$$, and $$\mu_0 (\leqslant \mu)$$. When in the slow phase, customers become impatient. That is, each customer, upon arrival, activates an individual timer, exponentially distributed with parameter $$\xi$$. If the system does not change its environment from 0 to 1 before the customer’s timer expires, the customer abandons the queue never to return.
We concentrate on deriving analytic solutions to the queue-length distributions. We derive, for each case of $$c$$, the corresponding probability generating function, and calculate the mean queue size. Several extreme cases are investigated and numerical results are presented.

### MSC:

 90B22 Queues and service in operations research
Full Text:

### References:

 [1] Altman, E.; Yechiali, U., Analysis of customers’ impatience in queues with server vacations, Queueing systems, 52, 261-279, (2006) · Zbl 1114.90015 [2] Altman, E.; Yechiali, U., Infinite-server queues with system’s additional tasks and impatient customers, Probability in the engineering and informational sciences, 22, 477-493, (2008) · Zbl 1228.60096 [3] Baccelli, F.; Boyer, P.; Hebuterne, G., Single-server queues with impatient customers, Advances in applied probability, 16, 887-905, (1984) · Zbl 0549.60091 [4] Baykal-Gursoy, M.; Xiao, W., Stochastic decomposition in $$M / M / \infty$$ queues with Markov modulated service rates, Queueing systems, 48, 75-88, (2004) · Zbl 1059.60093 [5] Boyce, W.; DiPrima, R., Elementary differential equations and boundary value problems, (2001), John Wiley and Sons · Zbl 0178.09001 [6] Boxma, O.J.; de Waal, P.R., Multiserver queues with impatient customers, Itc, 14, 743-756, (1994) [7] Bright, L.; Taylor, P.G., Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic models, 11, 3, 497-525, (1995) · Zbl 0837.60081 [8] Daley, D.J., General customer impatience in the queue $$\mathit{GI} / G / 1$$, Journal of applied probability, 2, 186-205, (1965) · Zbl 0134.14402 [9] Feller, W., An introduction to probability theory and its applications, (1957), Wiley New York · Zbl 0138.10207 [10] Gans, N.; Koole, G.; Mandelbaum, A., Telephone call centers: tutorial, review and research prospects, Manufacturing and service operations management, 5, 79-141, (2003) [11] V. Gupta, M. Harchol-Balter, A. Scheller-Wolf, U. Yechiali, Fundamental characteristics of queues with fluctuating load, in: Proceedings of the ACM SIGMETRICS, vol. 34, 2006, pp. 203-215. [12] S.P. Martin, I. Mitrani, Job transfers between unreliable servers, Presented in 2nd Madrid Conference on Queueing Theory, July 3-7, 2006. [13] Neuts, M.F., Matrix-geometric solutions in stochastic models – an algorithmic approach, (1981), Johns Hopkins Baltimore · Zbl 0469.60002 [14] O’Cinneide, C.A.; Purdue, P., The $$M / M / \infty$$ queue in a random environment, Journal of applied probability, 23, 175-184, (1986) · Zbl 0589.60076 [15] Palm, C., Methods of judging the annoyance caused by congestion, Tele, 4, 189-208, (1953) [16] Palm, C., Research on telephone traffic carried by full availability groups, Tele, 1, 107, (1957), English translation of results first published in 1946 in Swedish in the same journal, which was then entitled Tekniska Meddelanden fran Kungle. Telegrfstyrelsen [17] N. Paz, U. Yechiali, A note on the $$M / M / \infty$$ queue in random environment, Technical report, Department of Statistics and Operations Research, Tel-Aviv University, October 2007. · Zbl 1309.90019 [18] Takacs, L., A single server queue with limited virtual waiting time, Journal of applied probability, 11, 612-617, (1974) · Zbl 0303.60098 [19] Van Houdt, B.; Lenin, R.B.; Blondia, C., Delay distribution of (im)patient customers in a discrete time $$D - \mathit{MAP} / \mathit{PH} / 1$$ queue with age-dependent service times, Queueing systems, 45, 59-73, (2003) · Zbl 1175.90116 [20] Yechiali, U., A queueing-type birth-and-death process defined on a continuous-time Markov chain, Operations research, 21, 604-609, (1973) · Zbl 0288.60090 [21] Yechiali, U., Queues with system disasters and impatient customers when system is down, Queueing systems, 56, 195-202, (2007) · Zbl 1124.60076 [22] Yechiali, U.; Naor, P., Queueing problems with heterogeneous arrivals and service, Operations research, 19, 722-734, (1971) · Zbl 0226.60107
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