Stability analysis of a two-station cascade queueing network

We consider a two-station cascade network, where the first station has Poisson input and the second station has renewal input, with i.i.d. service times at both stations. The following partial interaction exists between stations: whenever the second station becomes empty while customers are awaiting service at the first one, one customer can jump to the second station to be served there immediately. However, the first station cannot assist the second one in the opposite case. For this system, we establish necessary and sufficient stability conditions of the basic workload process, using a regenerative method. An extension of the basic model, including a multiserver first station, a different service time distribution for customers jumping from station 1 to station 2, and an arbitrary threshold d1≥1 on the queue-size at station 1 allowing jumps to station 2, are also treated.

[1]  J. Dai.  On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models , 1995 .

[2]  Jim Dai.  Stability of open multiclass queueing networks via fluid models , 1995 .

[3]  Alexander L. Stolyar,et al.  Scheduling Flexible Servers with Convex Delay Costs: Heavy-Traffic Optimality of the Generalized cµ-Rule , 2004, Oper. Res..

[4]  Sigrún Andradóttir,et al.  Dynamic Server Allocation for Queueing Networks with Flexible Servers , 2003, Oper. Res..

[5]  Karl Sigman,et al.  One-Dependent Regenerative Processes and Queues in Continuous Time , 1990, Math. Oper. Res..

[6]  Alexander L. Stolyar,et al.  Control of systems with flexible multi-server pools: a shadow routing approach , 2010, Queueing Syst. Theory Appl..

[7]  Hong Chen.  Fluid Approximations and Stability of Multiclass Queueing Networks: Work-Conserving Disciplines , 1995 .

[8]  Evsey Morozov.  The tightness in the ergodic analysis of regenerative queueing processes , 1997, Queueing Syst. Theory Appl..

[9]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[10]  J. Walrand,et al.  Sufficient conditions for stability of longest-queue-first scheduling: second-order properties using fluid limits , 2006, Advances in Applied Probability.

[11]  D. McDonald,et al.  Large deviations of a modified Jackson network: Stability and rough asymptotics , 2005, math/0503487.

[12]  Elena Yudovina,et al.  Stochastic networks , 1995, Physics Subject Headings (PhySH).

[13]  HYUN-SOO AHN.  OPTIMAL CONTROL OF A FLEXIBLE SERVER , 2004 .

[14]  Onno J. Boxma,et al.  On queues with service and interarrival times depending on waiting times , 2007, Queueing Syst. Theory Appl..

[15]  Mahvareh Ahghari,et al.  Benefits of cross-training in a skill-based routing contact center with priority queues and impatient customers , 2009 .

[16]  Walter L. Smith,et al.  Regenerative stochastic processes , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[17]  Bara Kim,et al.  Stability of join-the-shortest-queue networks , 2007, Queueing Syst. Theory Appl..

[18]  S. Asmussen,et al.  Applied Probability and Queues , 1989 .

[19]  Evsey Morozov.  Weak Regeneration in Modeling of Queueing Processes , 2004, Queueing Syst. Theory Appl..

[20]  Z. Wang,et al.  Performance of service policies in a specialized service system with parallel servers , 2008, Ann. Oper. Res..

[21]  J. Michael Harrison,et al.  Heavy traffic resource pooling in parallel‐server systems , 1999, Queueing Syst. Theory Appl..

[22]  Atma Prakash Lalchandani.  SOME LIMIT THEOREMS IN QUEUEING THEORY , 1967 .

[23]  E. Morozov.  Instability of Nonhomogeneous Queueing Networks , 2002 .

[24]  R. J. Williams,et al.  Dynamic scheduling of a system with two parallel servers: asymptotic policy in heavy traffic , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[25]  E. Morozov,et al.  Stability analysis of regenerative queueing systems , 2009 .

[26]  Ronald W. Wolff,et al.  A Review of Regenerative Processes , 1993, SIAM Rev..

[27]  Tolga Tezcan,et al.  Stability analysis of N-model systems under a static priority rule , 2012, Queueing Systems.

[28]  S. Foss,et al.  AN OVERVIEW OF SOME STOCHASTIC STABILITY METHODS( Network Design, Control and Optimization) , 2004 .

[29]  Ronald J. Williams,et al.  Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy , 2001 .

[30]  J. Christopher Beck,et al.  An extended queueing control model for facilities with front room and back room operations and mixed-skilled workers , 2009, Eur. J. Oper. Res..

[31]  Yi-Chun Tsai,et al.  Dynamic server assignment policies for assembly‐type queues with flexible servers , 2008 .

[32]  Mark S. Squillante,et al.  Cycle stealing under immediate dispatch task assignment , 2003, SPAA '03.

[33]  D. Iglehart,et al.  Multiple channel queues in heavy traffic. I , 1970, Advances in Applied Probability.

[34]  R. J. Williams,et al.  Dynamic Scheduling of a System with Two Parallel Servers in Heavy Traac with Complete Resource Pooling: Asymptotic Optimality of a Continuous Review Threshold Policy 1 , 1999 .

[35]  D. Down,et al.  Stability of Queueing Networks , 1994 .

[36]  Saligrama R. Agnihothri,et al.  Workforce cross-training decisions in field service systems with two job types , 2003, J. Oper. Res. Soc..

[37]  Douglas G. Down,et al.  Dynamic load balancing in parallel queueing systems: Stability and optimal control , 2006, Eur. J. Oper. Res..

[38]  Wallace J. Hopp,et al.  Pooling strategies for call center agent cross-training , 2009 .

[39]  Evsey Morozov,et al.  A multiserver retrial queue: regenerative stability analysis , 2007, Queueing Syst. Theory Appl..

[40]  Ronald W. Wolff,et al.  A note on the existence of regeneration times , 1994, Journal of Applied Probability.

[41]  Zhang Hanqin,et al.  MULTIPLE CHANNEL QUEUES IN HEAVY TRAFFIC , 1990 .

[42]  Tolga Tezcan,et al.  Dynamic Control of N-Systems with Many Servers: Asymptotic Optimality of a Static Priority Policy in Heavy Traffic , 2010, Oper. Res..

[43]  Evsey Morozov,et al.  Stability analysis of GI/GI/c/K retrial queue with constant retrial rate , 2010, Math. Methods Oper. Res..

[44]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[45]  A. Borovkov.  Some Limit Theorems in the Theory of Mass Service, II Multiple Channels Systems , 1965 .

[46]  Sigrún Andradóttir,et al.  Robustness of efficient server assignment policies to service time distributions in finite‐buffered lines , 2010 .