Stochastic Decomposition in M/M/∞ Queues with Markov Modulated Service Rates

Motivated by the need to study transportation systems in which incidents cause traffic to slow down, we consider an M/M/∞ queueing system subject to random interruptions of exponentially distributed durations. System breakdowns, where none of the servers work, as well as partial failures, where all servers work with lower efficiency, are investigated. In both cases, it is shown that the number of customers present in the system in equilibrium is the sum of two independent random variables. One of these is the number of customers present in an ordinary M/M/∞ queue without interruptions.

[1]  Robert B. Cooper.  Queues served in cyclic order: Waiting times , 1970, Bell Syst. Tech. J..

[2]  U. Yechiali,et al.  Utilization of idle time in an M/G/1 queueing system Management Science 22 , 1975 .

[3]  J. George Shanthikumar.  On Stochastic Decomposition in M/G/1 Type Queues with Generalized Server Vacations , 1988, Oper. Res..

[4]  Robert B. Cooper,et al.  Stochastic Decompositions in the M/G/1 Queue with Generalized Vacations , 1985, Oper. Res..

[5]  P. Naor,et al.  Some Queuing Problems with the Service Station Subject to Breakdown , 1963 .

[6]  L. Slater,et al.  Confluent Hypergeometric Functions , 1961 .

[7]  Naishuo Tian,et al.  Analysis on queueing systems with synchronous vacations of partial servers , 2003, Perform. Evaluation.

[8]  William G. Marchal,et al.  State Dependence in M/G/1 Server-Vacation Models , 1988, Oper. Res..

[9]  J. Keilson.  Queues Subject to Service Interruption , 1962 .

[10]  Remo Guidieri.  Res , 1995, RES: Anthropology and Aesthetics.

[11]  Tayfur Altiok.  Queues with group arrivals and exhaustive service discipline , 1987, Queueing Syst. Theory Appl..

[12]  James MacGregor Smith,et al.  Modeling Vehicular Traffic Flow using M/G/C/C State Dependent Queueing Models , 1997, Transp. Sci..

[13]  D. Gaver.  A Waiting Line with Interrupted Service, Including Priorities , 1962 .

[14]  D. L. Fitzgerald.  Tricomi and Kummer functions in occurrence, waiting time and exceedance statistics , 2002 .

[15]  L. Christie,et al.  Queuing with Preemptive Priorities or with Breakdown , 1958 .

[16]  Xiuli Chao,et al.  Analysis of multi-server queues with station and server vacations , 1998, Eur. J. Oper. Res..

[17]  K. D. Tocher,et al.  Confluent Hypergeometric Functions , 1960 .

[18]  Julian Keilson,et al.  The matrix M/M/∞ system: retrial models and Markov modulated sources , 1993 .

[19]  M. Yadin,et al.  Queueing Systems with a Removable Service Station , 1963 .

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

[21]  S. W. Fuhrmann.  Technical Note - A Note on the M/G/1 Queue with Server Vacations , 1984, Oper. Res..

[22]  B. Avi-Itzhak,et al.  A Many-Server Queue with Service Interruptions , 1968, Oper. Res..