Overflow traffic in closed queueing models
Muhammad El-Taha,
Department of Mathematics and Statistics, University of Southern Maine
John Heath,
Department of Computer Science, University of Southern Maine
Abstract
We consider a "generalized" birth death process that represents a
multiserver (closed or open) overflow queueing system
with n primary servers and c - n secondary servers. An arrival to the system joins a server of the primary group, if available,
otherwise it overflows to the secondary group.
If all servers are busy, arrivals are queued, provided the queue buffer is not full,
and served
as servers become free. Arrivals that find the queue buffer
full are lost.
Our overflow model differs from models in the open literature in that it
combines state dependent arrival rates with group
dependent service rates.
We present a general formulation that allows a simple
derivation of the joint stationary probabilities of i busy primary
servers and j busy secondary severs.
This main result easily lends itself to an efficient iterative
algorithm to evaluate the joint probabilities.
We apply our basic theorem to produce new results for several overflow models.
A distinctive feature of our approach is that it uses
transform-free analysis.