The waiting line models or the queuing problem is identified by the
presence of a group of customers who arrive randomly to receive some
service. The customer upon arrival may be attended to immediately or
may have to wait until the server is free. The service time required to
serve the customers is also a statistical variable. This methodology can be
applied in the field of business (banks, booking counters), industries
(servicing of machines), government (railway or post-office counters),
transportation (airport, harbor) and everyday life (elevators, restaurants,
and doctor’s chamber).
The waiting line or queuing models are basically relevant to service
oriented organisations and suggest ways and means to improve the
efficiency of the service. An improvement of service level is always
possible by increasing the number of employees. Apart from increasing
the cost an immediate consequence of such a step is unutilized or idle time of the
servers. In addition, it is unrealistic to assume that a large-scale increase in staff is
possible in an organisation.
Queuing methodology indicates the optimal usage of existing manpower and
other resources to improve the service. It can also indicate the cost implications if
the existing service facility has to be improved by adding more servers.
The relationship between queuing and service rates can be
diagrammatically illustrated using the cost curves shown in Figure.
At a slow service rate, queues build up and the cost of queuing increases.
An ideal service unit will minimise the operating cost of the entire system.
Many real-life situations in which study of queuing theory can provide
solution to waiting line problems are listed below:
Situation Customers Service Facilities
Petrol pumps (stations) Automobiles Pumps
Hospital Patients Doctors/Nurses/Rooms
Airport Aircraft Runways
Post office Letters Sorting system
Job interviews Applicants Interviewers
Cargo Trucks Loader/unloaders
Workshop Machines/Cars Mechanics/Floor space
Factory Employees Cafeteria/Punching Machine
CHARACTERISTICS OF A WAITING LINE OR A
QUEUEING MODEL
A queuing system can be described by the following components:
Arrival
The statistical pattern of the arrival can be indicated through the
probability distribution of the number of arrivals in an interval. This is a
discrete random variable. Alternatively, the probability distribution of the
time between successive arrivals (known as inter-arrival time) can also
be studied to ascertain the stochastic aspect of the problem. This variable
is continuous in nature.
The probability distribution of the arrival pattern can be identified through
analysis of past data. The discrete random variable indicating the number
of arrivals in a time interval and the continuous random variable
indicating the time between two successive arrivals (interarrival time)
are obviously inter related. A remarkable result in this context is that if the
number of arrivals follows a Poisson distribution, the corresponding
interarrival time follows an exponential distribution. This property is
frequently used to derive elegant results on queuing problems.
Service
The time taken by a server to complete service is known as service time.
The service time is a statistical variable and can be studied either as the
number of services completed in a given period of time or the completion
period of a service. The data on actual service time should be analyzed to
find out the probability distribution of service time.
Server
The service may be offered through a single server such as a ticket counter
or through several channels such as a train arriving in a station with
several platforms.
Sometimes the service is to be carried out sequentially through several
phases known as multiphase service. In government, the papers move
through a number of phases in terms of official hierarchy till they arrive at
the appropriate level where a decision can be taken.
Time spent in the queuing system
The time spent by a customer in a queuing system is the sum of waiting
time before service and the service time.
Queue discipline
The queue discipline indicates the order in which members of the queue
are selected for service. It is most frequently assumed that the customers
are served on a first' come first serve basis. This is commonly referred to
as FIFO (first in, first out) system. Occasionally, a certain group of
customers receive priority in service over others even if they arrive late.
This is commonly referred to as priority queue; the queue discipline does
not always take into account the order of arrival. The server chooses one
of the customers to offer service at random. Such a system is known as
service in random order (SIRO). While allotting an item with high demand
and limited supply such as a test match cricketer share of a public limited
company, SIRO is the only possible way of offering service when it is not
possible to identify the order of arrival.
Size of a population
The collection of potential customers may be very large or of a moderate
size. In a railway booking counter the total number of potential passengers
is so large that although theoretically finite it can be regarded as infinity
for all practical purposes. The assumption of infinite population is very
convenient for analysing a queuing model. However, this assumption is
not valid where the customer group is represented by a few looms in a
spinning mill that require operator facility from time to time. If the
population size is finite then the analysis of queuing model becomes more
involved.
Maximum size of a queue
Sometimes only a finite number of customers are allowed to stay in the
system although the total number of customers in the population may or
may not be finite. For example, a doctor may make appointments with k
patients in a day. If the number of patients asking for appointment exceeds
k, they are not allowed to join the queue. Thus, although the size of the
population is infinite, the maximum number permissible in the system
is k.
