# Queueing Systems with Daily Cycles and Stochastic Demand with Uncertain Parameters

### Andrew M. Ross

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## Abstract

Telecommunications systems see large daily cycles in demand, and when
they are designed the demand they will face is uncertain. We look at
three typical telephone-related queueing systems from the
Internet-access and call-center industries.

Our first model considers the cost of providing service via
sub-contractors. We present a cost model for taking Internet dial-up
traffic that varies by time-of-day and splitting it between two large
modem banks, one of which charges by the hour, the other charges for
the peak. To study if the possible savings are enough to make the
effort worthwhile, we formulate a ``perfect information'' Integer
Program that is equivalent to a network flow problem. In the
stochastic case, we use a Normal approximation, and develop a
square-root-type rule to set the ceiling in the homogeneous case. We
also use simulation to determine an optimal ceiling when we cannot
route individual calls precisely.

In our second model, we estimate the number of modems and
characteristics of the opposing traffic stream in a partially observed
Erlang loss system (individual modem bank). Using detailed sample-path
data, we construct a Maximum Likelihood Estimator that makes good use
of the data, but is slow to evaluate. As an alternative, we present
an estimation system based on traffic data summarized by hour. This
estimation system is much faster, and tends to produce good lower
bounds on the size of the system and competing traffic.

In our third model, we consider designing a call center system when we
cannot predict the amount of traffic accurately. We try to decide on
a number of servers and number of extra buffer spaces (number of calls
that can wait on-hold), given a prior distribution on the input
traffic. The systems are designed to satisfy upper bounds on the
probabilities of blocking or delay. For infinite-buffer systems, we
find that a previously published approximation tends to under-estimate
the needed number of servers. For finite-buffer systems with known
traffic, we adapt the approximation to handle the trade-off between
adding servers and adding buffers.