MAP/M/c and M/PH/c queues with constant impatience times

This paper considers stationary MAP/M/c and M/PH/c queues with constant impatience times. In those queues, waiting customers leave the system without receiving their services if their elapsed waiting times exceed a predefined deterministic threshold. For the MAP/M/c queue with constant impatience times, Choi et al. (Math Oper Res 29:309–325, 2004) derive the virtual waiting time distribution, from which the loss probability and the actual waiting time distribution are obtained. We first refine their result for the virtual waiting time and then derive the stationary queue length distribution. We also discuss the computational procedure for performance measures of interest. Next we consider the stationary M/PH/c queue with constant impatience times and derive the loss probability, the waiting time distribution, and the queue length distribution. Some numerical results are also provided.

[1]  P. H. Brill,et al.  Level Crossings in Point Processes Applied to Queues: Single-Server Case , 1977, Oper. Res..

[2]  R. Haugen,et al.  Queueing Systems with Stochastic Time out , 1980, IEEE Trans. Commun..

[3]  Tetsuya Takine,et al.  Multi-class M/PH/1 queues with deterministic impatience times , 2017 .

[4]  Tetsuya Takine,et al.  A note on the virtual waiting time in the stationary PH/M/c+D queue , 2015, Journal of Applied Probability.

[5]  B. Conolly.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1991 .

[6]  E. Seneta.  Non-negative Matrices and Markov Chains , 2008 .

[7]  Nam Kyoo Boots,et al.  AnM/M/c queue with impatient customers , 1999 .

[8]  Ali Movaghar-Rahimabadi.  On queueing with customer impatience until the beginning of service , 1998, Queueing Syst. Theory Appl..

[9]  Marcel F. Neuts,et al.  The first two moment matrices of the counts for the Markovian arrival process , 1992 .

[10]  Bara Kim,et al.  A single server queue with Markov modulated service rates and impatient customers , 2015, Perform. Evaluation.

[11]  Peter G. Taylor,et al.  Invariant measures for quasi-birth-and-death processes , 1998 .

[12]  Andreas Brandt,et al.  On the M(n)/M(n)/s Queue with Impatient Calls , 1999, Perform. Evaluation.

[13]  Paul J. Schweitzer,et al.  Stochastic Models, an Algorithmic Approach , by Henk C. Tijms (Chichester: Wiley, 1994), 375 pages, paperback. , 1996, Probability in the Engineering and Informational Sciences.

[14]  M. Miyazawa.  The derivation of invariance relations in complex queueing systems with stationary inputs , 1983 .

[15]  Jerim Kim,et al.  M/PH/1 QUEUE WITH DETERMINISTIC IMPATIENCE TIME , 2013 .

[16]  Jacob Cohen,et al.  On up- and downcrossings , 1977, Journal of Applied Probability.

[17]  Shaler Stidham,et al.  The Relation between Customer and Time Averages in Queues , 1980, Oper. Res..

[18]  D. Y. Barrer.  Queuing with Impatient Customers and Ordered Service , 1957 .

[19]  L. M. Pedroso,et al.  Letter to the Editor: Computing multiple integrals involving matrix exponentials , 2008 .

[20]  Chun-Hua Guo,et al.  Convergence Analysis of the Latouche-Ramaswami Algorithm for Null Recurrent Quasi-Birth-Death Processes , 2001, SIAM J. Matrix Anal. Appl..

[21]  Tatsuya Suda,et al.  Mean Waiting Times in Nonpreemptive Priority Queues with Markovian Arrival and i.i.d. Service Processes , 1994, Perform. Evaluation.

[22]  Peter W. Glynn,et al.  Computing Poisson probabilities , 1988, CACM.

[23]  Ronald W. Wolff,et al.  The equality of the virtual delay and attained waiting time distributions , 1990 .

[24]  Bhaskar Sengupta,et al.  An invariance relationship for the G/G/1 queue , 1989, Advances in Applied Probability.

[25]  Bara Kim,et al.  MAP/M/c Queue with Constant Impatient Time , 2004, Math. Oper. Res..

[26]  Tetsuya Takine,et al.  Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions , 2001, Queueing Syst. Theory Appl..

[27]  Colm Art O'Cinneide,et al.  Representations for matrix-geometric and matrix-exponential steady-state distributions with applications to many-server queues , 1998 .

[28]  Valerie Isham,et al.  Non‐Negative Matrices and Markov Chains , 1983 .

[29]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[30]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[31]  Anders Rygh Swensen,et al.  On a GI/M/c Queue with Bounded Waiting Times , 1986, Oper. Res..

[32]  Søren Asmussen,et al.  Calculation of the Steady State Waiting Time Distribution in GI/PH/c and MAP/PH/c Queues , 2001, Queueing Syst. Theory Appl..

[33]  Vaidyanathan Ramaswami,et al.  A logarithmic reduction algorithm for quasi-birth-death processes , 1993, Journal of Applied Probability.

[34]  M. Neuts,et al.  A single-server queue with server vacations and a class of non-renewal arrival processes , 1990, Advances in Applied Probability.

[35]  L. M. Pedroso,et al.  Computing multiple integrals involving matrix exponentials , 2007 .

[36]  C. Loan.  Computing integrals involving the matrix exponential , 1978 .