Abstract
We consider an Mt/M/1 queueing system with impatient customers and a single vacation, assuming the customers’ impatience is due to the server’s vacation. In the context of non-stationary sinusoidal modeling, this paper introduces systems with exponential service times and periodic (sinusoidal) Poisson arrival processes. We studied a novel analysis of an Mt/M/1 model including simultaneous vacations and impatient customers alongside the relative amplitude changes. In addition, the pointwise stationary approximation has been computed by integrating over time the formula for the stationary performance measure with the arrival rate that applies at each point in time. The time-dependent probability generating functions and the corresponding steady-state results have been obtained explicitly. We focus on five performance measures: the expected number of customers waiting in the queue during vacation, the expected customer waiting time in the queue during vacation, the probability of the server being busy, the probability of the server being on vacation and the probability of customers’ impatience. Finally, to evaluate the performance measure of queue length, we have conducted a sensitivity analysis by running a simulation for a specific set of parameters.
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References
Altman, E. and Yechiali, U. (2006). Analysis of customers’ impatience in queues with server vacation, Spr. Sc. Bus. Media Queueing Systems, 52, 261-279, https://doi.org/10.1007/s11134-006-6134-x.
Dequan, Y., Wuyi, Y., Saffer, Z. and Chen. X. (2014). Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, Journal of Industrial and Management Optimization, 89–112, https://doi.org/10.3934/jimo.2014.10.89.
Edie, L. C. (1954). Traffic Delays at Toll Booths, Journal of the Operations Research Society of America, 2(2), https://doi.org/10.1287/opre.2.2.107.
Green, V. L. and Kolesar, P. J. (1991). The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals, Management Science, 37, 84-97, https://doi.org/10.1287/mnsc.37.1.84.
Green, V. L. and Kolesar, P. J. (1995). On the Accuracy of the Simple Peak Hour Approximation for Markovian Queues, Management Sciences, 41(8),1353-1370, https://doi.org/10.1287/mnsc.41.8.1353.
Green, V. L., Kolesar, P., J. and Svoronos, A. (1991). Some Effects of Non-stationarity on Multiserver Markovian Queueing Systems, Operations Rese., 39(3), 502-511, https://doi.org/10.1287/opre.39.3.502
Green, V. L., Kolesar, P., J. and Soares, J. (1999). Improving the SIPP Approach for Staffing that Have Cyclic Demands, Ins., for Operations Research, 49 (4), 549–564.
Green, V. L., Kolesar, P., J. and Whitt, W. (2007). Coping with Time-Varying Demand When Setting Staffing Requirements for a Service System, Columbia University, Production and Operation Management, 16(1), 13 – 39, https://doi.org/10.1111/j.1937-5956.2007.tb00164.x.
Heyman, D. P. and Whitt, W. (1984). The Asymptotic Behavior of Queues with Time-Varying Arrival Rates, Journal of Applied Probability, 21(1), 143-156, https://doi.org/10.2307/3213672.
Ingolfsson, A., Akhmetshina, E., Budge, S., Li, Y. and Wu, X. (2007). A Survey and Experimental Comparison of Service-Level-Approximation Methods for non-stationary Mt/M/s Queueing Systems with Exhaustive Discipline. Informs Journal on Computing, 19(2), 201–214. https://doi.org/10.1287/ijoc.1050.0157.
Kapodistria, S. (2011). The M/M/1 queue with synch. abandonments. Que. Systems, 68, 79–109.
Kolesar, P. (1984). Stalking the Endangered CAT: A Queueing Analysis of Congestion at Automatic Teller Machines, Graduate School of Business Uris Hall Columbia University, Management Sciences, New York, 14(6), 16- 26, https://doi.org/10.1287/inte.14.6.16.
Koopman, B. O. (1972). Air Terminal Queues under Time Dependent Conditions, Institute for Operations Research, 20(6),1089-1114, https://doi.org/10.1287/opre.20.6.1089.
Larson, R. (1972). Urban Police Patrol Analysis, MIT Press, Cambridge Mass, NCJ No, 9138.
Mahajan S. (2020), Non-stationarity and Abandonment in Markovian Queues with Application to Call Centers, American Journal of Operations Management and Information Systems 5(4), 74-83.
Margolius, B. H. (1999). A sample path analysis of the Mt/M/c queue, Que. Systems 31, 59–93.
Michael, H. R., Shmuel S. O. (1979). A Closure Approximation for the Non-stationary M/M/s Queue, Management Sciences 25(6): 522-534, https://doi.org/10.1287/mnsc.25.6.522.
Mieke, D., Inneke, V., N. (2011). Setting staffing levels in systems with time-varying demand, the context of an emergency department Research Center for Operations Management, Department of Decision Sciences and Information, Management, K. U. Leuven, Faculty of Business and Economics, https://doi.org/10.2139/ssrn.1749366.
Pant, A P. and Ghimire. R. P. (2015). Mt/M/1 Queueing System with Sinusoidal Arrival Rate, Journal of the Institute of Engineering, 11(1), 120-127.
Rolski, T. (1981). Queues with non-stationary input stream: Ross’s conjecture, 13 (3), 603- 618, https://doi.org/10.2307/1426787.
Rolski, T. (1986). Upper Bounds for Single Server Queues with Doubly Stochastic Poisson Arrival, Mathematics of Operations Research, in USA, 11 (3), https://doi.org/10.1287/moor.11.3.442.
Ross, S. M. (1978). Average Delay in Queues with Non-stationary Poisson Arrivals, J. Appl. Prob., 15 602- 609, https://doi.org/10.2307/3213122.
Roubos, A. (2015). Evaluation of service level approximations in call centers Approximations of the non-stationary Mt/M/s queue, VU University Faculty of Sciences De Boelelaan 1081a 1081 HV Amsterdam the Netherlands.
Shortle, J. F., Thompson, J. M., Gross, D., Harris, C. M. (2018). Fundamentals of Queueing Theory, 5th Edition, John Wiley and Sons, Inc., Hoboken, NJ 07030, USA
Whitt, W. (1991). The Pointwise Stationary Approximation for Mt/Mt/s Queues is Asymptotically Correct as the Rates Increase, AT & T Bell Laborat., Murray Hill, New Jersey 07974-2070.
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We kindly acknowledge the Allameh Tabataba’i University for supporting this research work.
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Yousefi, A., Pourtaheri, R. & Rad, M.R.S. Analysis of the Mt/M/1 Queueing System with Impatient Customers and Single Vacation. Sankhya B (2024). https://doi.org/10.1007/s13571-024-00326-y
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DOI: https://doi.org/10.1007/s13571-024-00326-y