Queuing Theory

Prof. Abhijeet Shivane

Introduction

• Queues (or waiting lines) is a common phenomenon in our everyday life.

• Queues are formed at the bus stands, traffic signals, telephone booths, enquiry windows, reservation counters, bank and post office counters.

• Queues are also formed in industrial units.• For example :- Wait at the tool crib to

obtain tools.

Introduction 02

• Queues need not be a physical line of customers

• It may be merely a list of customers, units, orders etc.

• Examples :-– Unconnected Telephone calls– Waiting list of passengers for a berth– Unscheduled orders in PPC dept– Untyped bunch of papers with typist

Introduction ..3

• In manufacturing excessive waiting time of WIP ( work in process) will mean longer manufacturing cycle , that is longer deliveries and higher investment in WIP, Idle time of workers, production loss etc.

Introduction …4

• Queues are formed when the current demand for a given service exceeds the capacity of the service facility to render service.

• One can shorten queues or prevent queues by employing additional service facilities but they may not be always desirable.

• The cost of additional service facilities reduces margin of profit while excessive waiting time results in loss of sale/loss of customers.

• Queuing theory gives mathematical treatment to waiting line problems

• It helps to strike an optimum balance between cost associated with waiting and cost associated with prevention of waiting.

Optimum balance may be arrived at either by :-

• Introducing more facilities at extra cost or

• Displacing less efficient service facilities by more efficient one or

• Changing the pattern of arrival of customers for service or

• Effective method improvements to reduce service time.

Approaches to solve a queuing problem

• Mathematical approach• Simulation approach • Mathematical approach :- Mathematical approach makes use of

probability distribution concept to represent arrival and service rates.

• Simulation approach:- Simulation approach is an iterative method wherein an approximate solution to the problem is obtained through simulated experiments based on random samples drawn from inter-arrival and service time distribution .

Elements of a Queuing system

• The simplest queue system consists of single queue and a single service facility

• It consists of – Arrival of customers– Waiting in queue – Being serviced and departure of the

customers

Elements of a Queue System

Input (Customers)

Waiting Line (Queue)

Server (Service Facility)

Input (Customers)

Output (Customers)

Basic Elements of a queuing system

• Input or arrival process:-

• Customers – workers, people, telephone calls, machines, WIP etc arrive at a service counter for service.

• Two important characteristics of the input process are it’s size and the pattern of arrival.

Basic Elements of a queuing system ..2

• Size represents the number of customers that arrive from time to time for service and later (pattern of arrival) suggests the distribution of these arrivals.

• The size of the input process is generally assumed to be infinite as this assumption facilitates analysis.

• The arrival may be at a constant rate or at random in accordance with some probability distribution.

• When the arrivals are completely random , they follow poisson’s distribution with mean arrival rate equal to average number of arrivals per unit time.

i) Input process

i) Input process :-• Arrival distribution• Inter arrival distribution• Mean arrival rate ( i.e. the average no. of

arrivals per unit time) represented by λ (Lambda)

• Mean Time between consecutive arrivals (i.e. time elapsed between first arrival and the next). This equals 1/ λ

ii) Waiting Line

• Waiting Time :- This implies the time that a customer spends in the queue before being taken up for service.– It equals the time that elapses between the

arrival of the customer and the commencement of the service to that customer.

ii) Waiting Line …2

• Service Time :- It is the time spent by the service facility to render service to a customer. It may be either constant or variable.

• Waiting Time in the system :- It is the time spent by the customer in the queue system. It equals waiting time plus service time.

• Queue Length :- It implies the no. of customers waiting in the queue.

• System Length :-It implies No. of customers in the queue plus those being serviced.

iii) Service discipline

• It represents the rule by which the next customer in the waiting line is selected for service.

• The most commonly used rule in a queue system is “First come first served”.

• This rule is widely practiced at the bus stand, booking counters, bank counters etc.

• Some times certain customers are given priority over others like express trains are given priority over passenger trains.

• Priorities tend to increase waiting time.

iv) Customer Behaviour

• A new customer may not join the queue because of his/her reluctance to wait.This trait is referred to as ‘balking’

• Only few of the new customers may join the queue and hence some of them may demand service on their behalf s well as on behalf of some other customers. This action of there is called ‘collusion’.

iv) Customer Behaviour …2

• A customer may join a queue , wait upto certain time and then leave the queue system without getting service. This action of customer is called as ‘reneging'. (breaking a promise)

• A customer may keep on switching from one queue to another. This happens when there are more than one service counter.The tendency so observed is called ‘Jockeying’.

V) Service facility

• Service facility represents servers – clerks, maintenance crew, machines which render service.

• To analyse a service facility both, number of servers and arrangement of servers ( in parallel or in series etc) need to be considered.

• There can be – Single queue :- single server.– Single queue :-multiple servers ( series arrangement)– Single queue :-multiple servers (parallel arrangement)– Multiple queue :-multiple servers

vi) System Output

• System output refers to rate at which customers are rendered service (i.e. rate at which customers leave the queue system after service.)

• System output is dictated by the service time required by the facility to render service and arrangement of the service facility.

vi) System Output ..2

• Service Time can be fixed or variable• Service time in most of the cases follow

exponential distribution. • Even though service time is variable yet average

no. of customers that can be served per unit time called service rate, can be obtained by constructing service timer frequency distribution.

• Service rate is represented by µ• Reciprocal of service rate is called service time.

(1/µ)

System Output ..2

• System output also depends on the arrangement of service facility system output.

• In a single queue single server arrangement equals service rate of service facility

• In a single queue multiple servers in series arrangement equals minimum of service rate of service facilities

• In a multiple queue multiple servers in parallel arrangement equals sum of the service rates of service facilities In a multiple queue multiple servers in series arrangement equals sum of the minimum service rates of service facilities

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