Elements of a queueing system

A one-stage queueing process is fully determined when six characteristics of the queueing system are known or specified. These elements are conventionally denoted by Kendall’s notation in the form of 𝐴/𝑆/𝑐/𝐵/𝐾/𝑆𝐷 (e.g., Bhat, 2015): interarrival time of customers (𝐴), service time of servers (𝑆), the number of parallel service channels (𝑐), the waiting room capacity, i.e., the number of buffers (𝐵), the population size of customers (𝐾) and the queue discipline (𝑆𝐷). They describe a queueing system and its behavior adequately. The interarrival time and the service time reflect the two sources of randomness; the other three elements reflect how the system is structured.

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3.3.2.1 Interarrival time 𝑨

Let us denote the beginning of the queueing process by 𝑇0. For a queueing

process with no customer inside the system at 𝑇0, it is also the beginning of the

arrival process. The incoming stream is a sequence of customers arriving for service. It can be described by the sequence of instants 𝑇1, 𝑇2, 𝑇3, . . . at which

the customers 1, 2, 3, . . . arrive. The arrival pattern is fully characterized by the sequence of times between successive arrivals, i.e., the interarrival times, defined as 𝐴𝑖 = 𝑇𝑖 −𝑇_𝑖−1, for 𝑖 ∈ _N, assuming 𝑇₀ = 0. For the modeling, it is necessary to specify the probability distribution(s) of the stochastic process 𝐴𝑖as well as

their stochastic dependence. If the probability distribution does not depend on time, the arrival pattern is stationary. The simplest arrival pattern is to have independent and identically distributed (i.i.d.) interarrival times 𝐴𝑖.

Alternatively, the arrival pattern can by characterized by the mean arrival rate, i.e., the average number of customers that arrive within a time unit. This characterization is useful when the arrival process is a Poisson process (see Section 3.3.3). In this case, the interarrival times are exponentially distributed with𝜆, which equals the mean arrival rate.

In most applications, arrivals do not happen simultaneously, and this is the mathematically simpler case. But it is possible for customers to arrive in groups (bulk or batch arrivals).

3.3.2.2 Service pattern (𝑺 and 𝒄)

The service pattern is described by the service time 𝑆 and the number of parallel servers 𝑐 together, representing how efficient the service can be delivered. Similar to the interarrival time, the service time is also a sequence of random variables 𝑆𝑖,

indicating the duration of the service for the 𝑖-th customer, ordered by the arrival. Much of the discussions regarding the interarrival time also applies to the service time, although the arrival process and the service process are generally assumed to be mutually independent. Most important is the probability distribution of the service times at each single server, which is i.i.d. in the simplest case. The service mechanisms of parallel channels are usually assumed to operate independently of each other. Likewise, the service rate at a single server is an

alternative characterization for independent parallel servers with i.i.d. service times. The number of servers that are able to provide the service independently in parallel is assumed to be time independent.

Generally, a queueing system is conceived in such a way that one customer is served at a time by a given server, but there are also situations where customers may be served simultaneously by the same server, such as passengers boarding a plane, a computer with parallel processors, and so on. If both arrival and service are bulked, then the entire process is purely parallel, corresponding to the parallel account in visual search literature.

3.3.2.3 Characteristics relevant to the structure (𝑩, 𝑲 and 𝑺𝑫)

The waiting room capacity 𝐵 refers to the maximum number of customers that are allowed to wait in the queue. In many applications, there is no obvious upper bound to the queue and arrived customers do not leave the queue before getting the service. In this case, the capacity considered as infinite. If the length of the queue has a limit, no further customer is allowed to enter when the queue is at its limit. The incoming customers are forced to leave without getting a service until there is space available again.

Likewise, the customer population 𝐾 is also a positive integer that can be seen as infinite if it is large enough or renewable. As mentioned in the last section, there are situations where the customer population is finite but each customer will enter the queueing system recurrently. The difference between a finite source with recurrent requests and an infinite source due to the renewability is that the arrival pattern of the customers from a finite source depends on how many customers are inside of (and consequently, also on the number of customers outside of) the system, whereas this dependence does not exist for infinite source1. In both cases, the queueing process is infinite in the sense that it goes on for

1_{An example for a finite source queue is a group of machines that require repair service}

when they become inoperative. When the defect happens to each machine independently and constantly, the more machines in the queue (waiting for repair or being repaired), the fewer machines are there that can become inoperative. Consequently, the arrival rate is lower when there are more customers in queue. In contrast, a post office in New York during the business hours shows an example for the cases in which a queue with an infinite source is an appropriate model.

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infinite time (as a stochastic process). In contrast, the queueing model of visual search should be truly finite in the sense that the queueing (search) process terminates to initiate a response.

The queue discipline 𝑆𝐷 refers to the way in which customers standing in line are selected for service. The simplest and most common queue discipline is first come first served (FCFS). There are other disciplines such as last come first served (as in the case of using an elevator), selected by priority and so on.

In general, if we treat the queuing system as a black box and observe only the incoming and outgoing stream, we may see a quite irregular pattern with large variations in the time intervals between departures. The behavior of the queue depends largely on the ratio of the arrival rate to the effective service rate (i.e., service rate of a single server multiplied by the number of servers). This ratio is called traffic intensity. A fundamental result of queueing theory regards how to use the traffic intensity as an indicator for the long-term behavior of the queue (Gross et al., 2008). If it is larger than one, then there are more customers arriving than the system can serve in a time unit. The queue will get longer and longer. If it equals one, a steady state still does not exist unless the queue is deterministic. If the traffic intensity is smaller than one, then the queue will reach a steady state eventually. But even in this case, some customers might be delayed by waiting in the line because randomness can lead to periods in which either too many servers are free or all servers are busy.

Figure 3.1 illustrates a clip of a single-line, multiserver queueing system at a specific time point. A server is getting free due to the departure of the 𝑗 + 2 customer and the next customer (who has arrived as the 𝑖-th at an earlier time) in line will be assigned to this server.