Analyzing priority queues with 3 classes using tree-like processes

In this paper we demonstrate how tree-like processes can be used to analyze a general class of priority queues with three service classes, creating a new methodology to study priority queues. The key result is that the operation of a 3-class priority queue can be mimicked by means of an alternate system that is composed of a single stack and queue. The evolution of this alternate system is reduced to a tree-like Markov process, the solution of which is realized through matrix analytic methods. The main performance measures, i.e., the queue length distributions and loss rates, are obtained from the steady state of the tree-like process through a censoring argument. The strength of our approach is demonstrated via a series of numerical examples.

[1]  Marcel F. Neuts,et al.  Markov chains with marked transitions , 1998 .

[2]  滝根 哲哉.  A Nonpreemptive Priority MAP/G/1 Queue with Two Classes of Customers , 1997 .

[3]  Dario Bini,et al.  Solving nonlinear matrix equations arising in Tree-Like stochastic processes , 2003 .

[4]  Marcel F. Neuts,et al.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1989 .

[5]  Attahiru Sule Alfa,et al.  Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue , 2003 .

[6]  Raymond W. Yeung,et al.  Matrix product-form solutions for Markov chains with a tree structure , 1994 .

[7]  Qi-Ming He,et al.  Queues with marked customers , 1996, Advances in Applied Probability.

[8]  Lajos Takács,et al.  Priority queues , 2019, The Art of Multiprocessor Programming.

[9]  Attahiru Sule Alfa,et al.  The quasi-birth-death type markov chain with a tree structure , 1999 .

[10]  A. Alfa.  Matrix‐geometric solution of discrete time MAP/PH/1 priority queue , 1998 .

[11]  Tetsuya Takine,et al.  The Nonpreemptive Priority MAP/G/1 Queue , 1999, Oper. Res..

[12]  Attahiru Sule Alfa,et al.  Advances in matrix-analytic methods for stochastic models , 1998 .

[13]  Douglas R. Miller.  Computation of Steady-State Probabilities for M/M/1 Priority Queues , 1981, Oper. Res..

[14]  C. Blondia,et al.  Tree Structured QBD Markov Chains and Tree‐Like QBD Processes , 2003 .

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

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

[17]  Fumiaki Machihara.  ON THE QUEUE WITH PH-MARKOV RENEWAL PREEMPTIONS , 1993 .

[18]  Tetsuya Takine,et al.  A generalization of the matrix M/G/l paradigm for Markov chains with a tree structure , 1995 .

[19]  Chris Blondia,et al.  Throughput of Q-Ary Splitting Algorithms for Contention Resolution in Communication Networks , 2004, Commun. Inf. Syst..

[20]  N. K. Jaiswal,et al.  Priority queues , 1968 .

[21]  Tetsuya Takine,et al.  The workload in the MAP/G/1 queue with state-dependent services: its application to a queue with preemptive resume priority , 1994 .