System Engineering 3(3+0)

LESSON 1. Introduction to queuing theory

QUEUEING THEORY is based on mathematical theories and deals with the problems arising due to flow of customers towards the service facility.

The waiting line models help the management in balancing between the cost associated with waiting and the cost of providing service. Thus, queuing or waiting line models can be applied in such situations where decisions have to be taken to minimize the waiting time with minimum investment cost.A flow of customers from in infinite/finite population towards the service facility forms. a queue (waiting line) on account of lack of capability to serve them all at a time. The queues may be of persons waiting at a doctor's clinic or at railway booking-office; these may be of machines waiting to be repaired or of ships in the harbor waiting to be unloaded or of letters arriving at a typist's desk. In the absence of a perfect balance between the service facilities and the customers, waiting is required either of the service facilities or for the customer's' arrival.

By the term 'customer' we mean the arriving unit that requires some service to be performed. The customer may be of persons, machines, vehicles, parts etc. Queues (waiting line) stand for a number of customers waiting to be serviced. The queue does not include the customer being serviced. The process or system that performs the services to the customer is termed by service channel or service facility.

 

QUEUEING SYSTEM

The mechanism of a queuing process is very simple. Customers arrive at service counter and are attended by one or more of the servers. As soon as a customer is served, he departs from the system. Thus a queuing system can be described as composed of customers arriving for service, waiting for service if it is not immediate, and if having waited for service, leaving the system after being served.

The detailed character station of a queuing system is defined by its characteristics discussed in the following section.

Queuing Model

It is a suitable model used to represent a service oriented problem, where customers arrive randomly to receive some service, the service time being also a random variable.

Arrival

The statistical pattern of the arrival can be indicated through the probability distribution of the number of the arrivals in an interval.

Service Time

The time taken by a server to complete service is known as service time.

Server

It is a mechanism through which service is offered.

Queue Discipline

It is the order in which the members of the queue are offered service.

Poisson Process

It is a probabilistic phenomenon where the number of arrivals in an interval of length t follows a Poisson distribution with parameter t, where  is the rate of arrival.

Queue

A group of items waiting to receive service, including those receiving the service, is known as queue.

Waiting time in queue

Time spent by a customer in the queue before being served.

Waiting time in the system

It is the total time spent by a customer in the system. It can be calculated as follows:

Waiting time in the system = Waiting time in queue + Service time

Queue length

Number of persons in the system at any time.

Average length of line

The number of customers in the queue per unit of time.

Average idle time

The average time for which the system remains idle.

FIFO

It is the first in first out queue discipline.

Bulk Arrivals

If more than one customer enter the system at an arrival event, it is known as bulk arrivals.

Components of Queuing System

1. Input Source: The input source generates customers for the service mechanism. The most important characteristic of the input source is its size. It may be either finite or infinite. Please note that the calculations are far easier for the infinite case, therefore, this assumption is often made even when the actual size is relatively large. If the population size is finite, then the analysis of queuing model becomes more involved.

   The statistical pattern by which calling units are generated over time must also be specified. It may be Poisson or Exponential probability distribution. Usually the source population is considered as unlimited.

2. Queue: It is characterized by the maximum permissible number of units that it can contain. Queues may be infinite or finite.

3. Service Discipline: It refers to the order in which members of the queue are selected for service. Frequently, the discipline is first come, first served.

 

Following are some other disciplines:

LIFO (Last In First Out)

SIRO (Service In Random Order)

Priority System

 

Service Mechanism: A specification of the service mechanism includes a description of time to complete a service and the number of customers who are satisfied at each service event. The service mechanism also prescribes the number and configuration of servers. If there is more than one service facility, the calling unit may receive service from a sequence of these. At a given facility, the unit enters one of the parallel service channels and is completely serviced by that server. Most elementary models assume one service facility with either one or a finite number of servers.The following figure shows the physical layout of service facilities.

 

Customer's Behaviour

1. Balking. A customer may not like to join the queue due to long waiting line.

2. Reneging. A customer may leave the queue after waiting for sometime due to impatience.

3. Collusion. Several customers may cooperate and only one of them may stand in the queue.

4. Jockeying. When there are a number of queues, a customer may move from one queue to another in hope of receiving the service quickly.

 

Server's Behaviour

• Failure. The service may be interrupted due to failure of a server (machinery).

• Changing service rates. A server may speed up or slow down, depending on the number of customers in the queue. For example, when the queue is long, a server may speed up in response to the pressure. On the contrary, it may slow down if the queue is very small.

• Batch processing. A server may service several customers simultaneously, a phenomenon known as batch processing.

 

Assumptions:

• The source population has infinite size.

• The inter-arrival time has an exponential probability distribution with a mean arrival rate of  l customer arrivals per unit time.

• There is no unusual customer behavior.

• The service discipline is FIFO.

• The service time has an exponential probability distribution with a mean service rate of m service completions per unit time.

• The mean arrival rate is less than the mean service rate, i.e., l < m.

• There is no unusual server behavior.