Analysis of a Queue with General Service Demands and Multiple Servers with Variable Service Capacities

We present the study of a non-classical discrete-time queuing system in which the customers each request a variable amount of service, called their “service demand”, from a system with multiple servers, each of which can provide a variable amount of service, called their “service capacity”, in each time slot. The service demands are independent from customer to customer and follow a general distribution, whereas the service capacities follow a phase-type distribution and are independent from server to server and from slot to slot. Since an exact analytical analysis for this general queuing system is infeasible, we propose several approximations for the key performance characteristics in this system such as the mean system content and the mean customer delay in steady state. The accuracy of each of these approximations is compared to simulations using several numerical examples.

[1]  A. Dudin,et al.  Self-Service System with Rating Dependent Arrivals , 2022, Mathematics.

[2]  Mor Harchol-Balter,et al.  WCFS: a new framework for analyzing multiserver systems , 2021, Queueing Systems.

[3]  M. Chaudhry,et al.  A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots , 2020, Methodology and Computing in Applied Probability.

[4]  Evsey Morozov,et al.  Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution , 2020, Mathematics.

[5]  Mohan L. Chaudhry,et al.  Analytically Simple and Computationally Efficient Results for the GIX/Geo/c Queues , 2019, Journal of Probability and Statistics.

[6]  Herwig Bruneel,et al.  Analysis of a queue with general service demands and correlated service capacities , 2018, Annals of Operations Research.

[7]  Herwig Bruneel,et al.  Analysis of discrete-time queues with general service demands and finite-support service capacities , 2017, Ann. Oper. Res..

[8]  Herwig Bruneel,et al.  Analysis of a discrete-time queue with general service demands and phase-type service capacities , 2017 .

[9]  Che Soong Kim,et al.  Hysteresis control by the number of active servers in queueing system MMAP/PH/N with priority service , 2016, Perform. Evaluation.

[10]  Herwig Bruneel,et al.  Discrete-time queues with variable service capacity: a basic model and its analysis , 2016, Ann. Oper. Res..

[11]  Attahiru Sule Alfa,et al.  Construction of Markov chains for discrete time MAP/PH/K queues , 2015, Perform. Evaluation.

[12]  Thomas Begin,et al.  Reduced complexity in M/Ph/c/N queues , 2014, Perform. Evaluation.

[13]  Daniel Wei-Chung Miao,et al.  Sample-path analysis of general arrival queueing systems with constant amount of work for all customers , 2014, Queueing Syst. Theory Appl..

[14]  Herwig Bruneel,et al.  The Discrete-Time Queue with Geometrically Distributed Service Capacities Revisited , 2013, ASMTA.

[15]  Bara Kim,et al.  Discrete-time multiserver queue with impatient customers , 2013 .

[16]  Herwig Bruneel,et al.  Analysis of a Discrete-Time Queue with Geometrically Distributed Service Capacities , 2012, ASMTA.

[17]  Hung Chen,et al.  ON THE VARIANCES OF SYSTEM SIZE AND SOJOURN TIME IN A DISCRETE-TIME DAR(1)/D/1 QUEUE , 2011, Probability in the Engineering and Informational Sciences.

[18]  Marc Moeneclaey,et al.  Calculation of delay characteristics for multiserver queues with constant service times , 2009, Eur. J. Oper. Res..

[19]  Johan van Leeuwaarden,et al.  On the application of Rouché's theorem in queueing theory , 2006, Oper. Res. Lett..

[20]  Natarajan Gautam,et al.  On Queues with Markov Modulated Service Rates , 2005, Queueing Syst. Theory Appl..

[21]  Dieter Fiems,et al.  A note on the discretization of Little's result , 2002, Oper. Res. Lett..

[22]  O. Boxma,et al.  The M/G/1 queue with two service speeds , 2001, Advances in Applied Probability.

[23]  Herwig Bruneel,et al.  Analysis of discrete-time multiserver queueing models with constant service times , 1994, Oper. Res. Lett..

[24]  Herwig Bruneel,et al.  Analyzing the discrete-timeG(G)/Geo/1 queue using complex contour integration , 1994, Queueing Syst. Theory Appl..

[25]  Herwig Bruneel,et al.  Discrete-time models for communication systems including ATM , 1992 .

[26]  S. Halfin,et al.  Steady-State Distribution for the Buffer Content of an $M /G/1$ Queue with Varying Service Rate , 1972 .

[27]  J. Little.  A Proof for the Queuing Formula: L = λW , 1961 .

[28]  D. Kendall.  Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain , 1953 .

[29]  Herwig Bruneel,et al.  Delay Analysis of a Queue with General Service Demands and Phase-Type Service Capacities , 2015, QTNA.

[30]  Herwig Bruneel,et al.  Discrete-time multiserver queues with geometric service times , 2004, Comput. Oper. Res..