Queuing System Characteristics

Choosing the appropriate queuing model is obviously an important factor to address the imbalance of the system appropriately. Model selection depends on the characteristics of the system that is researched. The main queuing model characteristics are: the population source; (i) number of servers (healthcare providers, i.e. nurses and physicians; (ii) arrival patterns and service patterns (iii); and queue discipline (iv).

58 The population source (i) for the ED is infinite. In an infinite source situation, patient arrival is

unrestricted and can greatly exceed system capacity at any given time. An infinite source exists when service or better the access to care, is unrestricted.

The capacity of queuing systems is determined by the treatment capacity of each server (ii), also known as a line or channel, and the number of servers being used. It is generally assumed that each channel can handle one customer at a time. Healthcare systems can be conceptualized as single-line or multiple-line, and may consist of phases (steps in a queuing system as shown in the conceptual model of the ED process).

Waiting lines occur random. Highly variable arrival and service patterns (iii) cause systems to be

(temporarily) overloaded or crowded. The ED is a typical example of erratic arrival patterns causing such variability. The arrival patterns might be different on mornings and afternoons, and even more so after GPs close in the evening. The most commonly used models assume that the patient arrival rate can be described by a Poisson distribution and that the time between arrivals, inter-arrival time, can be described by a negative exponential distribution. Service, or treatment, to the arriving patients is another element that exhibits variability. Because of the varying nature or illnesses of the patients, the time required for clinical attention (service/treatment times) varies from patient to patient. Service rate and service times are also interchangeably, so that the Poisson distribution can characterize the service rate.

The Poisson and the negative exponential distribution are alternate ways of presenting the same information. If service time is exponential, then the service rate is Poisson. Further, if the customer arrival rate is Poisson, then the inter-arrival rate (the time between arrivals) is exponential.

Queue discipline (iv) refers to the order in which customers are processed. The assumption that service is provided on a first-come, first-served basis is probably the most commonly encountered rule. First- come first-served, which is seen in many businesses, has special adaptations in healthcare queue discipline and includes the critical first rule (based on the triage codes as mentioned in chapter 2) at the ED. Queuing models are identified by their characteristics. From a methods perspective, a nomenclature of A/B/C/D/E is used to describe them. Table 5.1 provides details for each component of the

nomenclature. The last two components, D and E, of the nomenclature are not used unless there is a specific waiting room capacity or a limited population of patients. Since infinite-patient-source models are the main focus in this study, the nomenclature, “D and E,” will be omitted from discussion.

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Table 5.1 Queuing model classification

A: Specification of arrival process, measured by inter-arrival time or arrival rate.

M: Negative exponential or Poisson distribution. D: Constant value.

K: Erlang distribution.

G: A general distribution with known mean and variance.

B: Specification of service process, measured by inter-service time or service rate.

M: Negative exponential or Poisson distribution. D: Constant value.

K: Erlang distribution.

G: A general distribution with known mean and variance.

C: Specification of number of servers—“s”.

D: Specification of queue or the maximum numbers allowed in a queuing system. E: Specification of customer population.

There are two commonly used infinite source models: 1. Single channel, M/M/1

2. Multiple channel, M/M/s >1

The “M” in the models stands for Markov. This will be explained later on. The “s” designates the number of channels (servers or healthcare providers). (74)

A queuing model needs input before calculation can be made. These models assume steady state conditions and a Poisson arrival rate. The most commonly used symbols in queuing models are shown in table 5.2.

Figure 5.2 Queuing model notation

λ μ 𝑳𝒒 L 𝑾𝒒 W ρ 1/ μ 𝑷𝟎 Arrival rate Service rate

Average number of patients waiting for service

Average number of patients in the system (waiting or being served) Average time patients wait in line

Average time patients spend in the system System utilization

Service time

Probability of zero patients in system

The simplest model represents a system that has one server called a single channel. The length of queue can be endless, just as the demand for medical services is. In M/M/1 queue models, arrival time cannot be greater than service time. Since there is only one server, the system can tolerate up to 100%

utilization. If arrival rates are more than service rates, then a multi-channel queue system is appropriate (M/M/s >1). Again patients arrival rate can be approximated by a Poisson distribution, and service time

60 by a negative exponential distribution, or Poisson service rate. The difference is with this system that multiple server are available in the system. Since the ED has more than 1 server (healthcare provider), the M/M/s>1 model is the appropriate one to use.

If we recall, the “M” in our model stands for Markov. A queuing model can be described as a continuous time Markov chain with a transition rate matrix on the state space {0,1,2,3,...}. This is the same

continuous time Markov chain as in a birth–death process. The state space diagram for this chain in figure 5.1 shows the transitions states in the M/M/s>1 queue. A process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history, hence independently from such history; that

is conditional on the present state of the system, its future and past states are independent. A Markov chain is a type of Markov process that has either discrete state space or discrete index set, often representing time as is the case in our study.

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5.2 Methods