Abstract
We consider a system comprised of two connected M/M/•/• type queues, where customers of one queue act as servers for the other queue. One queue, Q 1, operates as a limited-buffer M/M/1/N−1 system. The other queue, Q 2, has an unlimited-buffer and receives service from the customers of Q 1. Such analytic models may represent applications like SETI@home, where idle computers of users are used to process data collected by space radio telescopes. Let L 1 denote the number of customers in Q 1. Then, two models are studied, distinguished by their service discipline in Q 2: In Model 1, Q 2 operates as an unlimited-buffer, single-server M/M/1/∞ queue with Poisson arrival rate λ 2 and dynamically changing service rate μ 2 L 1. In Model 2, Q 2 operates as a multi-server M/M/L 1/∞ queue with varying number of servers, L 1, each serving at a Poisson rate of μ 2.
We analyze both models and derive the Probability Generating Functions of the system’s steady-state probabilities. We then calculate the mean total number of customers present in each queue. Extreme cases are indicated.
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References
Arazi, A., Ben-Jacob, E., Yechiali, U.: Controlling an oscillating Jackson-type network having state-dependent service rates. Math. Methods Oper. Res. 62, 453–466 (2005)
Avi-Itzhak, B., Mitrani, I.L.: A many-server queue with service interruptions. Oper. Res. 16, 628–638 (1968)
Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA, Alexandria (1999)
Levy, Y., Yechiali, U.: An M/M/s Queue with servers’ vacations. Can. J. Oper. Res. Inf. Process. 14, 153–163 (1976)
Mahabhashyam, S.R., Gautam, N.: On queues with Markov modulated service rates. Queueing Syst. 51, 89–113 (2005)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models—An Algorithmic Approach. Johns Hopkins, Baltimore (1981)
Nunez Queija, R., Boxma, O.J.: Analysis of a multi-server queueing model of ABR. J. Appl. Math. Stoch. Anal. 11, 339–354 (1998)
Ozawa, T.: Sojourn time distribution in the queue defined by a general QBD process. Queueing Syst. 53, 203–211 (2006)
Perel, E., Yechiali, U.: 2-queue systems where customers of one queue serve the customers of the other queue. Technical Report, Department of Statistics and Operations Research, Tel Aviv University (2007)
Takine, T.: Single-server queues with Markov-modulated arrivals and service speed. Queueing Syst. 49, 7–22 (2005)
Yechiali, U.: A queueing-type birth-and-death process defined on a continuous Markov chain. Oper. Res. 21, 604–609 (1973)
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Perel, E., Yechiali, U. Queues where customers of one queue act as servers of the other queue. Queueing Syst 60, 271–288 (2008). https://doi.org/10.1007/s11134-008-9097-2
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DOI: https://doi.org/10.1007/s11134-008-9097-2