Uncertainty in the Demand for Service - Faculty of Industrial ...

3 Uncertain Arrival Rate
When designing service systems, we usually use stochastic models such as Erlang models, which
capture the uncertainty associated with the arrivals and service time. In addition, we assume
that these models’ parameters, such as the arrival rate and mean service time, are known. Then,
we analyze our system, computing performance measures and making staffing decisions under
these assumptions. However, very often the parameters’ values are not known with certainty.
This parameter uncertainty may lead to incorrect decisions. If we underestimate the true arrival
rate then we will not staff enough servers, which may be critical in an emergency medical service,
and if we overestimate the true arrival rate, we will provide better service than required, at the
cost of paying too many salaries.
Henderson [8] and Steckley et al. [11] consider two legitimate interpretations of the term ”uncertainty
of the arrival rate”. The interpretations differ in terms of whether the arrival rate
is randomly varying, or simply unknown. According to the first interpretation, on any given
day, the arrival process is modeled as a Poisson process with a random time-dependent rate
functions. The assumption of second interpretation is that the arrival rate function does not
vary from day to day, but it is not known with certainty. The main performance measure that
the authors of [8] the and [11] consider is the fraction of customers that wait in queue at most
τ seconds. They claim that the two systems can not be analyzed in the same way and show,
for each case, how to approximate this performance measure using steady-state approximations
and sketch how to estimate it using simulation .
For data to be generated by a Poisson distribution, the mean and the variance should be approximately
the same. However, in many practical cases, the data show a variance that significantly
higher than the mean. In order to account for this overdispersion, Koole and Jongbloed [9]
propose to model the arrival rate over [0,T ] by a Poisson mixture model in the form
λ(t) = λ0(t) · Z, 0 ≤ t ≤ T.
This model takes into consideration two sources of variability as follows. The scalar function
λ0(t) is such that T
0 λ0(u)du = 1. The value λ0(t) · ∆t is approximately the relative part of
expected arrivals during (t, t + ∆t) out of all arrivals, and it captures the predictable variability
in the arrival rate (time variations of expected arrival rate), e.g. peak demand of calls on
Mondays between 10:00 to 11:00, on a regular basis. In other words, the predictable variability
related to the shape of the arrival rate function. The second variable Z is a random variable that
captures stochastic variability: it describes the total number of arrivals to the service system,
i.e. describes the fluctuations around the time-dependent expected arrival rate. This model is
an example for the first interpretation of uncertainty, given by [8] and [11].
Every service manager must also take into consideration costs and how they are affected by
staffing decisions. Whitt [12] proposes methods for staffing a single-class call center with uncertain
arrival rate and uncertain staffing due to employee absenteeism while his goal is to maximize
the expected net return. He uses a deterministic fluid approximation and a numerical algorithm
for the conditional performance measures of M/M/N+N queue, and shows that for large arrival
rates the fluid model become very accurate and the exact numerical solution adds only little
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