Basic Queuing Theory
In this chapter we introduce basic elements from queuing theory. This chapter is written for those with a deeper (theoretical) interest in queuing theory. The reader can also skip this part and use it for reference purposes only.
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Basic Queuing Theory
In this chapter we introduce basic elements from queuing theory. This chapter is written for those with a deeper (theoretical) interest in queuing theory. The reader can also skip this part and use it for reference purposes only. The majority of the text in this chapter appeared previously in [13].
3.1 Some General Queuing Concepts A queue can generally be characterized by its arrival and service processes, the number of servers, and the service discipline (Fig. 3.1). The arrival process is specified by a probability distribution that has an arrival rate associated with it, which is usually the mean number of patients who arrives during a time unit (e.g., minutes, hours, or days). A common choice for the probabilistic arrival process is the Poisson process, in which the inter-arrival times of patients are independent and exponentially distributed. The service process specifies the service requirements of patients, again using a probability distribution with associated service rate. A common choice is the exponential distribution, which is convenient for obtaining analytical tractable results. The number of servers in a healthcare setting may represent the number of doctors at an outpatient clinic, the number of MRI scanners at a diagnostic department, and so on. The service discipline specifies how incoming patients are served. The most common discipline is first come first serve (FCFS), where patients are served in order of arrival. Some patients may have priority over other patients. This can be such that the service of a lower priority patient is interrupted when a higher priority patient arrives (preemptive priority), or the service of the lower priority patient is finished first (non-preemptive priority).
M.E. Zonderland, Appointment Planning in Outpatient Clinics and Diagnostic Facilities, SpringerBriefs in Health Care Management and Economics, DOI 10.1007/978-1-4899-7451-8 3, © The Author 2014
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3 Basic Queuing Theory Service process Waiting room Arrival process
Departure process
Fig. 3.1 A simple queue
3.1.1 Performance Measures Typical measures for the performance of a system include the mean sojourn time, E[W ], the mean time that a patient spends in the queue and in service. The sojourn time is a random variable as it is determined by the stochastic arrival and service processes. The mean waiting time, E[W q ], gives the mean time a patient spends in the queue waiting for service. How E[W ] and E[W q ] are calculated depends, among other things, on the choice for the arrival and service processes, and is given for several basic queues in Sect. 3.2. Kendall’s Notation All queues in this book are described using the so-called Kendall notation: A/B/s, where A denotes the arrival process, B denotes the service process, and s is the number of servers. There are several extensions to this notation, see, for example, [10]. Clearly, there are many distinctive cases of queues: M/M/1: The single-server queue with Poisson arrivals and exponential service times. The M stands for th
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