 
Summary: Asymptotic overflow probabilities for multiple source
models
16 JanKees C.W. van Ommeren
16 October 31, 2000
21 Abstract In this paper we consider a communication system in discrete time with a finite
buffer where input is received from multiple on/off sources. The number of sources may be
finite or infinite. The on times of sources are independent and geometrically distributed
with mean 1/(1  p); in case the number of sources is finite (say S) the off times are
geometrically distributed with mean (S/µ)1/(1p) and in case the number of sources is
infinite, the number of sources that become active per time unit has a Poisson distribution
with mean µ. The output rate of the system is 1 packet per unit time; the input rate of an
active source is 1 packet per unit time too so that if one source is active, the buffer contents
remains constant. In this paper we concentrate on the overflow probability of the buffer
at an arbitrary epoch and the probability that a packet is lost (both in steady state); in
particular, we derive asymptotic approximations for these probabilities. We assume that
µ/(1  p) < 1.
23 Introduction
25 Model and notation
27 S, the number of sources (possibly infinite); Ni, the number of sources that become active
during the time slot (i1, i]; Ri, the number of sources that are active in the two consecutive
