Markov Processes with Constant Transition Rates of Size One & Two

Abstract

The two classical single server queueing systems, M/M/1 and M/M/1/N are generalized to allow constant transition rates of size two in addition to the standard constant transitions rates of size one. In terms of the queueing models, these new systems each allow customers to arrive or be served instantly in pairs as well as individually. The steady state distributions are explicitly determined and a condition for the existence of a steady state distribution is established in the infinite-state space case. Assuming that a steady state condition prevails, the canonical performance measures are determined. Expressions for the average number of customers in either system or queue are derived. Formulae for the average waiting time that a customer spends in each system or queue are also developed.

Key Words:

• Single server queue
• Batch queueing systems
• Markov processes with transition steps of size one and two
• Performance measures
• Bulk queue
• M/M/1
• M/M/1/N
• Steady state distributions