Queuing Theory

The queuing solution is suitable for solving a service-oriented problem where the customer arrives randomly to avail the service. In this service, time is a random variable. In a typical queuing situ­ation, customers arrive to avail the service at a service system, enter a waiting line, receive service and then leave. The queuing model is relevant in service-oriented industry such as logistics, trans­portation, shipping, hospitality and banking. The key elements of the process are:

Source Population. Normally, all sources involve a finite or limited number of customers. In a telephone system there is a large source population. When there are many customers, usually more than 100, the source can be treated as if it were infinite in size and the number of poten­tial customers will influence the arrival behaviour of customers for the service system.

Arrival Process. The arrival process describes the behaviour in which customers reach the ser­vice system. Customers may arrive in batches (e.g., family) or individually. Customers may also arrive on a scheduled basis (e.g., appointment with the dentist). Arrival process is measured either by arrival rate numbers per hour or inter-arrival (e.g., each visiting every 5 minutes). When service is provided on a scheduled basis, the arrival rate or inter-arrival time is fixed. In unscheduled situations, however, customers arrive in a random pattern. The random pattern in most queuing situations follows the Poisson distribution.

Waiting Line. Customers do not physically form a queue, but the queue is formed in the book­ing at their arrival in the system. The important factors to consider in a queuing system are size or capacity of the waiting area (customers may turn away if it is full), queue length (customer may refuse to join if it is too long) and queue organization.

Queue Discipline. The method by which customers from the waiting area are selected for service is referred to as the queue discipline. The following are queuing disciplines that can be followed:

  • First-in-first-out (FIFO)
  • Last-in-first-out (LIFO)
  • Priority scheme

Service Process. The simplest case is a single service facility, while others may consist of mul­tiple servers in one stage or multiple stage servers. The following are some of the examples:

  • Single server, single stage
  • Multiple servers, single stage
  • Multiple parallel, non-identical servers, single stage
  • Single server, multiple stages
  • Multiple server, multiple stages

Regardless of the design configuration, it will take time to perform the service at each server. There are two ways to describe this service process:

  1. Service Rate—the number of customers served per unit of time, for example, 30 per hour.
  2. Service time—the time taken to serve a customer, for example, 2 minutes per service.

Service time may be constant (e.g., machine processing) or may fluctuate (e.g., the checkout in a supermarket) within some range of value. We can use probability distribution (e.g., negative exponential distribution) to describe the service process if it is fluctuating.

Departure. Most customers may return to the queuing system after servicing, while others may never return again.

There are several measurements that should be considered when measuring the performance of a queuing system. However, the average value of the following measurements for a system in a steady state needs to be calculated:

λ: Average arrival rate

u: Average service rate

p: System utilization

LS: Average number of customers in the queuing system

Lq: Average number of customers in the waiting line

WS: Average time a customer spends in the system

Wq: Average time a customer spends in the waiting line

Pn: Probability of there being n customers in the queuing system.

There are two types of queuing systems, e.g. transient state (where probability of the number of customers in the system depends on time) and steady state (where probability of the number of visiting customers in the system is independent of time).

A single queue single server model is represented as (M/M/l),


M = Poisson arrival rate

M = Exponential service time

l = Single server

The assumptions here are: unlimited customers, unlimited waiting area, first-come-first-served, single server, arrival rate follows a Poisson distribution, and service time follows a negative expo­nential distribution. The formula calculations are:

For example, a chemical company distributes its products by trucks loaded at its only load­ing station. The tankers of both the company and the contractor are used for this purpose. It was found that on an average every 5 minutes, one truck arrived and the average loading time is 3 min­utes. Fifty per cent trucks belong to contractors. Using the queuing model and by making certain assumptions, one can find out the probability that a truck has to wait, waiting time of the truck, expected waiting time for the contractor’s truck before loading and so on.

Source: Sople V.V (2013), Logistics Management, Pearson Education India; Third edition.

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