# Accession Number:

## AD0681010

# Title:

## TWO SERVERS IN SERIES, STUDIED IN TERMS OF A MARKOV RENEWAL BRANCHING PROCESS.

# Descriptive Note:

## Technical rept.,

# Corporate Author:

## PURDUE UNIV LAFAYETTE IND DEPT OF STATISTICS

# Personal Author(s):

# Report Date:

## 1968-10-01

# Pagination or Media Count:

## 69.0

# 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 MG1 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 MG1 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

# Descriptors:

# Subject Categories:

- Operations Research