TWO SERVERS IN SERIES, STUDIED IN TERMS OF A MARKOV RENEWAL BRANCHING PROCESS.
Abstract
The paper discusses the transient and limiting behavior of a system of queues, consisting of two service units in tandem and 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 statedependent, death process. The analysis of the timedependence 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' one is led to consider functional iterates and a matrix analog of Takacs' functional equation for the transform of the distribution of the busy period in the M/G/1 model. Joint distribution of the queuelengths in units I and II and its marginal and limiting distributions are discussed. A final section is devoted to an informal discussion on how the numerical analysis of this system of queues may be organized. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1968
- Accession Number
- AD0681010
Entities
People
- Marcel F. Neuts
Organizations
- Purdue University