## 7.2 Characteristics of Queuing Systems

The key elements of queuing systems are customers and servers.

• The term customer can refer to people, machines, trucks, airplanes etc etc. Anything that arrive at a facility and requires service.

• The term server can refer to receptionist, repair personnel, runways in airport, washing machines etc etc. Any resource that provides the requested service.

Below we describe the elements of queuing systems in more details.

### 7.2.1 The Calling Population

The population of potential customers, referred to as the calling population, will be assumed to be infinite, even though the number of potential customers is actually finite. When the population of potential customers is large this assumption is innocuous and actually can simplify the model. This is especially true when we believe that at any given time the number of customers being served or waiting for service is a small proportion of the whole population.

The assumption of an infinite population is such that the rate of arrival of customers is not affected by the number of customers that have already joined the queuing system. In addition, this will usually entail that the rate of arrival is constant throughout time.

### 7.2.2 System Capacity

In many queuing systems there is a limit to the number of customers that may be in the waiting line or system. An arriving customer who finds the system full does not enter but returns immediately to the calling population. However, there are other systems that may simply have an infinite capacity. As we will see later, when a system has a limited capacity, a distinction is made between the arrival rate (i.e. the number of arrivals per time unit) and the effective arrival rate (the number who arrive and enter the system per time unit).

### 7.2.3 The Arrival Process

The arrival process for infinite population models is usually characterized in terms of inter-arrival times of successive customers. Arrivals may occur at scheduled times or random times. When at random times, the inter-arrival times are usually characterized by a probability distribution. In addition, customers may arrive one at a time or in batches. The batch may be of constant size or of random size.

The most important model, and the only one we will consider, for random arrivals is the Poisson arrival process. If $$A_n$$ represents the inter-arrival time between customer $$n-1$$ and customer $$n$$, then for a Poisson arrival process $$A_n$$ is exponentially distributed with mean $$1/\lambda$$ per time unite. The arrival rate is $$\lambda$$ customers per time unit. The number of arrivals in a time interval of length $$t$$ has the Poisson distribution with mean $$\lambda t$$ customers.

### 7.2.4 Queue Behavior and Queue Discipline

Queue behavior refers to the actions of customers while in a queue waiting for service to begin. In some situation, there is a possibility that incoming customers will balk, renege, or jockey (move from one line to another if they think they have chosen a slow line).

Queue discipline refers to the logical ordering of customers in a queue and determines which customer will be chosen for service when a server becomes free. Common queue disciplines include first-in-first-out (FIFO), last-in-first-out (LIFO), service in random order (SIRO) etc. Notice that a FIFO queue discipline implies that services begin in the same order as arrivals, but that customers could leave the system in a different order because of different length service times.

### 7.2.5 Service Times and Service Mechanism

The service times of successive arrivals are denoted by $$S_1,S_2,\dots$$. They may be constant or of random duration. In the latter case $$\{S_1,S_2,S_3,...\}$$ is usually characterized as a sequence of independent and identically distributed random variables. The Exponential, Normal etc. are often used to model service times. Sometimes services are identically distributed for all customers of a given type or class or priority, whereas customers of different types might have completely different service-time distributions. In addition in some systems service times depend upon the time of the day or upon the length of the waiting line.

A queuing system consists of a number of service counters and interconnecting queues. Each service center consists of some number of server, $$c$$, working in parallel; that is, upon getting to the head of the line, a customer takes the first available server. Parallel service mechanisms are either single server ($$c=1$$), multiple server ($$1<c<\infty$$), or unlimited servers $$(c=\infty)$$.