Location of single-server immobile facilities subject to a loss constraint

Abstract

Waiting may be unacceptable, even a short time, at a facility providing a service involving medical or other emergencies. Hence, it is appropriate to locate such facilities so that the rate at which users are lost is limited. Each facility will here be modelled as an M/E_r/m/N queueing system subject to a loss restriction constraint and the single-server case (m=1) will be treated in detail. Introduction of the Erlang distribution for service times allows a better fit of the model to actual values of both mean and variance than do currently available models that use an exponential distribution. Location of facilities will be such that the average travel time to a facility is minimized. It is shown how a deterministic constraint, equivalent to the loss constraint, can be generated resulting in an integer linear program, and values of a parameter ρ_c which facilitates this linearization are tabulated for various values of r, N and service level demanded. Numerical experiments are performed including an application loosely related to the location of neonatal clinics in the Municipality of Rio de Janeiro. Finally, there is a discussion of how further improved modelling of the service time distribution might be effected.

Keywords:

• location
• queues
• loss
• Erlang distribution

^† Sadly RD Galvão passed away before this paper appeared in print.

^† Sadly RD Galvão passed away before this paper appeared in print.

Acknowledgements

The first named author wishes to acknowledge support from EPSRC for this research via Grant EP/C51436X, and the second author the support from CNPq, the Brazilian National Research Council, via Grant 478996/2006-3. Thanks are also due to anonymous referees for their very helpful comments.