On the Timing of the Peak Mean and Variance for the Number of Customers in an M(t)/M(t)/1 Queueing System

Revised October 1994This paper examines the time lag between the peak in the arrival rate and the peaks in the mean and variance for the number of customers in an M(t)/M(t)/1l system. We establish a necessary condition for the time at which the peak in the mean is achieved. In cases in which system utilization exceeds one during some period, we show that the peak in the mean occurs after the end of this period

Modeling the M(t)/M/s(t) Queue with an Exhaustive Discipline

The M/M/s queueing model assumes arrival rates, service rates, and number of servers to be constant in time. It is straightforward to extend the model to allow a time-varying arrival rate and a time-varying service rate, and the resulting models can be solved numerically using standard solvers for ordinary differential equations. The extension to a time-varying number of servers requires consideration of the behavior of a server that is providing service when scheduled to leave. Past work has (implicitly or explicitly) assumed a pre-emptive discipline, where the customer rejoins the queue. We demonstrate how to model an exhaustive discipline that is often more realistic, where servers complete their current service before leaving. We discuss how to compute the virtual waiting time distribution for this model and outline its extension to incorporate customer abandonments

Abstract Maximum Availability Models for Selecting Ambulance Station and Vehicle Locations: A Critique

Several researchers have employed the notion of reliability of coverage to extend the set covering and maximal coverage models. We discuss the suitability of these models in general for choosing ambulance station or vehicle locations. Then, we discuss one particular model (the queueing maximum availability location problem) in greater detail and describe difficulties encountered in applying it using realistic data. Acknowledgments This work has been partly supported by Discovery grants 25481 and 203534 from the Natura

Ambulance location for maximum survival

This article proposes new location models for emergency medical service stations. The models are generated by incorporating a survival function into existing covering models. A survival function is a monotonically decreasing function of the response time of an emergency medical service (EMS) vehicle to a patient that returns the probability of survival for the patient. The survival function allows for the calculation of tangible outcome measures—the expected number of survivors in case of cardiac arrests. The survival-maximizing location models are better suited for EMS location than the covering models which do not adequately differentiate between consequences of different response times. We demonstrate empirically the superiority of the survival-maximizing models using data from the Edmonton EMS system

• Appendix D: Proof of Theorem 1 • Appendix E: Sensitivity to Service Time Distribution • Appendix F: Additional Computational Results

Two main streams of literature are relevant to the problem of considering server unavailability in emergency response systems. The first is that on the development of analytical models that allow for the calculation of measures related to server availability. The second is that related to location models for emergency service systems that incorporate such measures. Table A-1 summarizes the methods that will be discussed here in terms of the assumptions made about the four areas of the system outlined in the paper. Note that for the first three characteristics, the column heading is the characteristic and for each model we state the assumption made about that characteristic, but for the last characteristic (server cooperation) we focus on a specific aspect of the characteristic (server dependence) and only provide information about that aspect within the table. The reason for this is that all of the models incorporate some information about server cooperation, typically in the form of a “closest available ambulance” dispatch rule, and the main differences between the models in terms of this characteristic are in the way that they model the server dependence aspect. Additionally, we have attempted to list the models in order of increasing realism, although given the variety of assumptions, in some cases the order is admittedly subjective. A major development in the first stream is the hypercube queueing model (Larson, 1974), whic