Two servers in series, studied in terms of a Markov renewal branching process
- 1 January 1970
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 2 (1), 110-149
- https://doi.org/10.2307/3518346
Abstract
This paper discusses the transient and limiting behavior of a system of queues, consisting of two service units in tandem in which the second unit has finite capacity. When the second unit reaches full capacity, a phenomenon termed “blocking” occurs. A wide class of rules to resolve blocking is defined and studied in a unified way.The input to the first unit is assumed to be Poisson, the service times in the first unit are independent with a general, common distribution. When the system is not blocked, the second unit releases its customers according to a state-dependent, death process.The analysis of the time-dependence relies heavily on several imbedded Markov renewal processes. In particular, the analog of the busy period for the M/G/1 queue is modeled here as a “Markov renewal branching process”. The study of this process requires the definition of a class of matrix functions which generalizes some classical definitions of matrix function. In terms of these “matrix functions” we are led to consider functional iterates and a matrix analog of Takács' functional equation for the transform of the distribution of the busy period in the M/G/1 model.We further discuss the joint distribution of the queue lengths in units I and II and its marginal and limiting distributions.A final section is devoted to an informal discussion on how the numerical analysis of this system of queues may be organized.Keywords
This publication has 13 references indexed in Scilit:
- The joint distribution of the virtual waitingtime and the residual busy period for the M/G/1 queueJournal of Applied Probability, 1968
- Two queues in series with a finite, intermediate waitingroomJournal of Applied Probability, 1968
- Stability of finite queue, tandem server systemsJournal of Applied Probability, 1967
- The single server queue with Poisson input and semi-Markov service timesJournal of Applied Probability, 1966
- A Sequence of Two Servers with No Intermediate QueueManagement Science, 1965
- Limit Theorems for Markov Renewal ProcessesThe Annals of Mathematical Statistics, 1964
- A Correction to “The Solution of Queueing and Inventory Models by Semi-Markov Processes”Journal of the Royal Statistical Society Series B: Statistical Methodology, 1963
- Markov Renewal Processes with Finitely Many StatesThe Annals of Mathematical Statistics, 1961
- Markov Renewal Processes: Definitions and Preliminary PropertiesThe Annals of Mathematical Statistics, 1961
- Regenerative stochastic processesProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1955