M.Sc Student | Shimrit Maman |
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Subject | Uncertainty in the Demand for Service: The Case of Call Centers and Emergency Departments |

Department | Department of Industrial Engineering and Management |

Supervisor | Full Professor Mandelbaum Avishai |

Full Thesis text |

A standard assumption in the modeling of open service systems postulates that the customers' arrival process is Poisson with a known parameter. More specifically, the prevalent approach is to assume known arrival rates for each basic time interval (say, half-hour). In practice, however, arrival data violates this assumption by exhibiting more variability than the one expected from the Poisson hypothesis.

We explain this "over-dispersion" by "natural" uncertainty of the arrival rate, which gives rise to a Poisson mixture model for the arrival process. Then we incorporate this mixture model into the M|M|n+G queue and analyze it asymptotically, in steady state. Our approach is motivated by the seminal paper of Halfin and Whitt, assuming that both the mean arrival rate λ and the number of servers converge jointly to infinity.

It turns out that system performance strongly depends on the
order of over-dispersion. Specifically, we analyze three regimes of
arrival-rate uncertainty: order λ, λ^{0.5} and, λ^{c},
0.5<c<1. The latter case is especially practical, since it seems to fit
the call center reality. For the first two regimes, we derive asymptotically
optimal staffing levels, in the sense of being the least that adhere to a pre-specified waiting probability. Key operational
performance measures are asymptotically calculated as well. In the practical
regime, 0.5<c<1, we consider staffing levels that are characterized by λ^{c}
safety-staffing: n=R+βR^{c}+o(R^{0.5}), where R=λE[S]
is the offered load and β is a quality-of-service parameter. When β>0,
this staffing level reflects the need for protection against variability that
exceeds that in the square-root staffing of the conventional QED (Halfin-Whitt)
regime. If β<0, on the other hand, less protection is needed.

An extensive numerical study, based on data from an Israeli
call center, validates the practical model (arrival-rate uncertainty of order
λ^{c}, 0.5<c<1, specifically c≈0.8) and relates it to
an alternative well-known Poisson mixture model with an underlying Gamma prior.
In an additional numerical study, based on data from an emergency department of
a hospital, we get that c is around 0.5.

Finally, our Poisson mixture model is incorporated into the
M_{t}|M|n+G
queue, giving rise to the model with a random time-varying arrival rate. The
above-derived staffing regimes then give rise to time-varying staffing rules
which stabilize the delay probability - in fact, this time-stable performance
matches that of a specific stationary queue, at all times, remarkably well.