Queueing Theory Ppt

Introduction

• Also called ‘Waiting Line’ Theory.

• Owes its development to AK Erlang – efforts to

analyse telephone traffic congestion with the aim of

satisfying the randomly arising demand for the

services of Copenhagen telephone system in 1909.

• The theory is applicable to situations where:-

a) ‘Customers’ arrive; at

b) Some service stations; for

c) Some service.

d) Wait (occasionally not); then

e) Leave the system after getting the service.

Introduction

• Examples of such ‘arrival’/ ‘departure’ problems:

(a) Customers waiting to deposit electricity bills

(b) Machines waiting to be repaired

(c) Aircraft waiting to land at an airport

(d) Patients in a hospital who need treatment

• Service stations in these situations are:

(a) Cash counter

(b) Repairmen

(c) Runway

(d) Doctor

Introduction

• Why do waiting lines develop? Because service is not rendered

immediately on arrival of customer at the service facility.

• Why?

- Lack of adequate service facility

- Inefficiency of the existing facility

• Therefore, to meet service demand

(a) Increase service capacity; and /or

(b) Raise capacity of existing capacity/facility, if possible.

• It is possible to build capacity to such high level that can

always meet peak demand with no queues.

• This would become uneconomical after a stage because

capacity will remain idle to varying degrees when there are no

or very few customers.

Introduction

• Manager’s/ Management job is to balance.

• Inefficient/poor service excessive waiting cost in terms

of:-

(a) Customer frustration

(b) Loss of goodwill in the long run

(c) Direct cost of idle employees (e.g. employees waiting at

the store for collection of tools).

• Conversely too high a service level high set-up cost

idle time for service stations.

• Goal of queuing modelling achievement of economic

balance between

(a) Cost of providing service

(b) Cost associated with wait

Waiting Time and Idle Time Costs

To solve a queuing problem, service facility must be manipulated so that an optimum balance is obtained between the cost of waiting time and the cost of idle time. Cost of waiting customers includes:-

(a) Indirect cost of lost business – people may go elsewhere, buy lesser than intended, do not come again in future

(b) Direct cost of idle equipment and people –

(i) Cost of truck drivers and equipment to be loaded/unloaded

(ii) Cost of ship anchored in high seas waiting to enter the dockyard

(iii) Cost of operating an airplane kept circling overhead waiting for clearance to land

Waiting Time and Idle Time Costs

Cost of lost business – not easy to assess.

For example, vehicle drivers wanting petrol will avoid petrol stations having long queues.

To determine how much business is lost due to the avoidance by customers is a difficult task – some type of experimentation or data collection would be required.

Cost of idle service facilities is the payment made to the servers (fixed cost + variable cost) engaged at the facilities for the period for which they remain idle.

Relationship between level of service and waiting time cost

Increased Service

Wait

ing

tim

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ost

Relationship between level of service and cost 0f providing service

Increased Service

Cost

of

pro

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Total Expected Cost of operating the Facility

Increased Service

Tota

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Total expected cost

S = Optimum service level

Cost of providing serviceWaiting time cost

General Structure of Queuing System

• Arrival Process• Arrival (Input) Distribution• Service (Output) Distribution• Service System• Maximum number of people allowed into the system• Calling source or population• Customer Behaviour

Population

Arrival Process

QueueService System

Customers leave

Elements of Queuing System

1. Arrival Process.

(a) According to source

(i) Finite

(ii) Infinite

(b) According to numbers

(iii) Individual, as in a bank

(ii) In groups, as in a ship

(c) According to time

(iv) Deterministic, as in automated processes

(v) Stochastic (Probabilistic), at random

ELEMENTS OF QUEUING SYSTEM

2. Arrival (or input) distribution.

• It represents the pattern in which customers arrive at a service facility.

• Arrivals can be characterized either by an average arrival rate or by an average arrival time.

• An average arrival rate denotes the average number of arrivals in a given interval of time – e.g. two customers per hour, four trucks per minute, two potholes per kilometer of road, five typing errors per printed page.

• In contrast, average arrival time denotes the average time between two successive arrivals – e.g. 30 minutes between customers, 0.25 minutes between trucks, 0.5 kilometers between potholes, 0.2 pages between typing errors.

• Important to remember that for queuing models, we typically use the average rate.

• Arrivals may be separated by equal intervals of time, unequal but definitely known intervals of time, or by unequal intervals of time whose probabilities are known; the last one is termed as random arrivals.

…..Cont’d


2. Arrival (or input) distribution.

• It turns out that in many real-world queuing problems, even when the arrivals are random, the number of arrivals per interval can be estimated using the Poisson Probability Distribution.

• Poisson distribution is applicable whenever the following assumptions are made:-

(a) The average arrival rate over a given interval is known.

(b) This average rate is the same for all equal-sized intervals.

(c) The actual number of arrivals in one interval has no bearing on the actual number of arrivals in another interval.

(d) There cannot be more than one arrival in an interval as the size of the interval approaches zero.

• The mean value of the arrival rate, i.e. average arrival rate, is represented by λ (Lambda).


• For a given average arrival rate, a discrete Poisson distribution can be established by using the formula

P(x) = (λx. e-λ)/x! for x=0,1,2,….. where,

x = number of arrivals per interval

P(x) = Probability of exactly ‘x’ arrivals in an interval#.

λ = average arrival rate (i.e. average number of arrivals per unit interval)

e = 2.71828 or 2.7183

{# - The intervals could be time or space (length/ distance).}

• Properties of Poisson Probability distribution (which is a discrete probability distribution)

(a) Probability of an occurrence is the same for any two

intervals# of equal length.

(b) Occurrence or non-occurrence in any interval# is independent of the occurrence or non-occurrence in any

other interval#.


(3) Service (or output) distribution.

• Service patterns are like arrival patterns in that they can be either constant, variable but known, or random (variable with only known probability).

• If the service time is constant, it takes the same amount of time to take care of each customer – e.g. processes in a soft drinks industry.

• However, more often than not, service times are randomly distributed.

• It turns out that in real-world queuing problems, such situations are well represented by exponential distribution.

• Poisson and exponential probability distributions are directly related to each other. If the number of arrivals follows a Poisson distribution, it turns out that the time between successive arrivals follows an exponential distribution.

• Processes that follow these distributions are commonly referred to as Markovian processes.

• Mean value of service rate is represented by μ.

….Cont’d


Service Distribution

• Just as we did with arrivals, we need to distinguish here between service rate and service time – while the service rate denotes the number of units served in a given interval of time, the service time denotes the length of time taken to actually perform the service.

• Although the exponential distribution estimates the probability of service times, the parameter used in this computation is the average rate.

• Exponential distribution can be established using the formula:

P(t) = e- μt (t≥0) where,

t = Service time

P(t) = probability that service time will be greater than t

μ = average service rate (i.e. average number of customers served per unit time

e = 2.7183

Expected time (mean of the exponential distribution), E(t) = 1/λ

Note : Mean of exponential distribution is the inverse of the mean of the Poisson distribution.


(4) Service System(a) Service Channels (b) Speed of Service (c) Service Discipline

(a) Service Channels :(i) Single Service facility with single queue


Arrivals

Queue

Service facility Customers leave

Single Server Single Queue E.g. Library counter

Service System (a) Service Channels :

(ii) Multiple parallel facility with single queue


Service facility

Queue Customers leave

Arrivals

Multiple Parallel Servers Single Queue E.g. Car Service Station


(iii) Multiple parallel facility with multiple queues

QueueService facility

Customers leave

Arrivals

Arrivals

Arrivals

Arrivals

Multiple Parallel Servers Multiple Queues E.g. Railway Reservation Counter



(iv) Service Facilities in Series

Arrivals

Queue Service facility

Customers leave

Multiple Servers in Series E.g. Machining of an item consisting of cutting, turning, grinding drilling and packaging

Queue Service facility


Service System

(b) Speed of Service:

Speed with which service can be provided can be expressed in two ways :-

(i) Service Rate: No. of customers served during a unit of time.

(ii) Service Time: Amount of time needed to serve a customer.

For e.g., if a cashier can serve 10 customers in an hour, then Service Rate is 10 customers per hour and Service Time is 6 minutes per customer.

(c) Service Discipline:

(i) First come-first served (FCFS) or

(ii) Last come-first served (LCFS) or Last in-First out (LIFO) basis

(iii) Service in random order (SIRO)

(iv) Priority Service E.g. Customers called according to some identifiable characteristics (say length of job), treatment of VIPs in preference to other patients in a hospital


(5) Maximum number of people allowed into the system

• Maximum number can be ‘finite’ or ‘infinite’.

• In some facilities, only a limited number of customers or

units are allowed into the system and new arriving

customers/ units are not allowed to join the system unless

the number becomes lesser than the limiting value.

• For example, computerized system – up to limiting value you

get into the system and till you are connected to a server, a

pre-recorded message comes to you that you are in the

queue. However, in case you dial when there are more

customers than the limiting value already in the system, you

will not be allowed into the system and you get an engaged

tone.


(6) Calling source or population

- Arrival pattern depends on the source that generates them

- If only few potential customers – calling source or population is called ‘finite’

- If large number of potential customers – called ‘infinite’.

- Another rule for categorizing source as finite or infinite –

(i) A finite source exists when an arrival affects the probability of arrival of potential future customers.

For Example: say, ‘M’ machines are running – if one machine breaks down, it becomes a customer and cannot generate another ‘call’ till repaired or serviced, thus affecting the probability of breakdown of (M-1) machines, i.e. rate of arrival of the remaining machines. Hence, the calling source is termed as finite.

(ii) An infinite source exists when arrival of one customer does not affect the rate of arrival of potential future customers.


(7) Customer Behaviour or Queuing Behaviour

This element is only applicable where human customers are

involved.

(a) Jockeying – i.e. going/moving from one queue to another

in the hope of reducing waiting time.

(b) Balking – i.e. not joining the queue at all because of the

anticipated long delay.

(c) Reneging – Leaving the queue and going away because

they have been waiting too long


An analysis of a given queuing system involves a study of its

different operating characteristics. This is done using queuing

models. Commonly used operating characteristics are :-

(a) Queue Length (Lq) – Average number of customers in the

queue waiting to get service. Long queues may indicate poor

service performance or inadequate service facilities. Small

queues may imply too much server capacity.

(b) System Length (Ls) – Average number of customers in the

system, i.e. both those waiting to be served and those being

served.

….Cont’d

OPERATING CHARACTERISTICS OF QUEUING SYSTEM

(c) Waiting Time in the Queue (Wq) – Average time that a

customer has to wait in the queue to get service. Long

waiting time is directly related to customer dissatisfaction

and potential loss of future revenues. Very less waiting time

may indicate too much service capacity and scope to reduce

service cost.

(d) Total Time in the System (Ws) – Average time that a

customer spends in the system, from entry in the system till

he leaves the system on completion of service (i.e. waiting

time + service time).

(e) Utilization factor (ρ) – It is the proportion of time that a

server actually spends with the customers, i.e. average

utilization. It is also called ‘traffic intensity’ or ‘clearing

ratio’ or ‘proportion of time that the server is busy’.

OPERATING CHARACTERISTICS OF QUEUING SYSTEM

Queuing Models

Assumption:

• Various models that we would now be seeing are based on the

assumption that the system operates under ‘equilibrium or

steady state’ conditions and is not in the ‘transient’ stage.

• For example, opening of a departmental store during a

“normal” day or during a “big sale” day represent radically

different initial conditions (small or non-existent initial queue

versus long initial queue) which will affect the operating

characteristics during the early part of the operating period.

• However, as time goes by, the system will eventually settle

down to its long run or steady state tendencies.

• Another example for settling down into the steady state

condition in the long run is the concept of “learning curves”.

• When an aircraft is newly purchased, initial servicing time of

the aircraft is very high.

• With practice, however, the time period will gradually reduce

and eventually settle down to an average steady state condition.

Transient Phase

Steady State Phase

Steady State Mean

Time

No o

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Queuing Models

Queuing Models - Deterministic

• Queuing models can be categorized as “Deterministic” or “Probabilistic”.

DETERMINISTIC QUEUING MODEL

1. If each customer or unit arrives at fixed or known intervals and the service time is known with certainty, the queuing model would be deterministic in nature.

2. Suppose customers come to a bank’s teller counter every 5 minutes. Interval between arrivals of any two successive customers is exactly 5 minutes.

(a) Suppose the teller takes exactly 5 minutes to serve a customer – here both arrival and service rates = 12 customers per hour. In such a situation, there will never be a queue and the teller will always be busy.


(b) Suppose that the teller can serve 15 customers per hour – consequence of this higher service rate would be that the teller would be busy 4/5th of the time and idle for 1/5th of the time i.e. he would take 4 minutes to serve a customer and wait for 1 minute for the next customer to arrive. Therefore, there would be no queue.

(c) If teller can serve 10 customers per hour – teller will always be busy and queue length will increase continuously without limit i.e. if service rate is lesser than the arrival rate, the service facility cannot cope with all the arrivals and eventually the system leads to an explosive situation. Such a problem can only be solved by providing additional service facility.

Symbolically, if

λ = arrival rate of customers per unit time;

μ = service rate of customers per unit time; then,

if λ > μ a waiting line will be formed that will increase

indefinitely; the service facility will always be busy; and the

service system will eventually fail.

if λ < or = μ there will be no waiting line; the

proportion of time the service will be idle is (1- λ/μ).

λ/μ = ρ (pronounced ‘Rho’), is also called ‘average utilization’

or ‘traffic intensity’ or ‘clearing ratio’ or ‘proportion of time

that the server is busy’. ρ is the proportion of time that the

system will be busy.


So, in other words, if ρ < 1, the system will work and if ρ > 1, the

system will eventually fail.

Note:- It has been proved mathematically that in a probabilistic

model, even if λ = μ (i.e. ρ =1) the system will eventually fail.

Such deterministic models are only applicable for highly

automated plants – e.g. soft drinks industry, pharmaceutical

industry. Where humans are involved, variable arrival rates and

service times are the more realistic assumptions.


PROBABILISTIC QUEUING MODELS

Kendall’s Notation for representing queuing models is (a/b/c) :

(d/e/f), where

a = Arrival distribution (or inter-arrival time)

b = Service distribution (or service time)

c = number of parallel service channels in the system

d = Service discipline

e = maximum number of customers allowed into the system

f = calling source or population

Queuing Models - Probabilistic

Symbols for ‘a’(Arrival distribution) and ‘b’(Departure distribution)

M = Markovian (Poisson) arrival or departure (or exponential inter-arrival or service time distribution)

Ek = Erlangian or Gamma inter-arrival or service time.

GI = General independent arrival distribution.

G = General departure distribution

D = Deterministic inter-arrival or service distribution.

Symbols for ‘c’(No. of service channels)

1 for single server and ‘c’ or ‘s’ or ‘k’ for multiple servers.

Symbols for ‘d’(Service discipline)

FCFS, LCFS, SIRO, etc.

Symbols for ‘e’(Max. no of customers allowed into the system)

N = finite number of customers allowed in the system

∞ = infinite number of customers allowed in the system

Symbols for ‘f’(Calling

source/population)

N = finite population

∞ = infinite population

Following conventional codes are generally used to replace the symbols:-

For example:- (M/M/1) : (FCFS/∞/∞) ; (M/M/1) : (SIRO/∞/∞) ; M/M/c) : (FCFS/N/∞)


Queuing Models - Probabilistic• We shall concentrate on the (M/M/1) : (FCFS/∞/∞) model

• In this model, the customers’ arrivals follow Poisson distribution while the service times are exponentially distributed. To recapitulate, the Poisson probability function,

f(x) = e-μ . μx/ x! where,

f(x) = probability of ‘x’ occurrences in an interval

μ = expected value or mean number of occurrences in an interval.

• Using this formula for our purpose, with average arrival rate equal to λ per period of time, then according to Poisson probability distribution, the probability that ‘n’ customers will arrive in the system during a given time interval ‘t’ is given by

(a) P(‘n’ arrivals in time ‘t’) = {e-λt.(λ.t)n/n! } for n = 0, 1, 2, …..

(b) Probability density function (pdf) of inter-arrival time ( i.e. time interval between two consecutive arrivals) = λ.e-λt

• (Note: λ is assumed to be constant over time and is independent of the number of units already serviced or served, queue length or any other random property of the queue).


Similarly for Service,

• Service time is the time required for completion of a service, i.e. it is the time interval between beginning of a service and its completion.

• The mean service rate, μ, is the number of customers served per unit of time (assuming the service to be continuous throughout the entire time unit), while the average service time, 1/μ, is the time required to serve one customer.

• Most common type of distribution used for service times is exponential distribution.

• It involves the probability of completion of a service.


Note :

1. Poisson distribution cannot be applied to servicing because of the possibility of the service facility remaining idle for some time (Poisson distribution assumes fixed time intervals of continuous servicing which can never be assured in all services and at all times).

2. μ is also assumed to be constant over time and independent of the number of units already serviced, queue length or any other random property of the system.

3. If λ is the arrival rate, the inter-arrival time is 1/λ, and if μ is the service rate, then the service time is 1/μ.


(a) P (‘n’ complete services in time ‘t’) = e-μt.(μ.t)n/n!

(b) Probability density function (pdf) of inter-service time (or time

between two consecutive services) = μ.e-μt

(c) Probability that customer shall be serviced in more than time

‘t’= e-μt

(d) P (no more than ‘t’ time period needed to service a customer) =

1 – e-μt

Problem 1

• On an average, 6 customers reach a booth every hour to make calls. Determine the probability that exactly 4 customers will reach in 30-minute period, assuming that arrivals follow Poisson distribution.

• SolutionGiven λ = 6 customers per hour t = 30 minutes = 0.5 hour n = 4 λt = 6*0.5 = 3

Probability of exactly 4 customers reaching in 30-minute period,

= e-λt.(λt)n/n! = e-3.34/4! = 0.168

Simple Problems

Simple Problems

Problem 2

• In a bank, 20 customers on the average are served by a cashier in an hour. If the service time has exponential distribution, what is the probability that –

(a) It will take more than 10 minutes to serve a customer?(b) A customer shall be free within 4 minutes?

• Solution

Given μ = 20 customers per hour t = 10 minutes = 10/60 = 1/6 hour(a) Probability that it will take more than 10 minutes to serve a customer

= e-μt = e-20*1/6 = 0.0357(b) In this case, t = 4 minutes = 1/15 hour ; μ = 20 customers per hour

Probability that a customer shall be free within 4 minutes = 1 – e-μt = 1 – e-20*1/15 = 0.736

Some performance measures/operating characteristics of general interest for evaluation of the performance of an existing queuing system and to design a new system in terms of the level of service a customer receives as well as the proper utilization of the service facilities are:-

1. Time-related questions for the customers.

(a) What is the average (or expected) time an arriving customer has to wait in the queue (denoted by Wq) before

being served?

(b) What is the average (or expected) time an arriving customer spends in the system (denoted by Ws) including

waiting and service? This data can be used to make economic comparison of alternative queuing systems.

Performance Measures of a Queuing System

2. Quantitative questions related to the number of customers.

(a) Expected number of customers who are in the queue (Queue length) for service denoted by Lq.

(b) Expected number of customers who are in the system either waiting in the queue or being served (denoted by Ls). The data can be used for finding the mean customer time spent in the system.



3. Questions involving value of time for both customers and servers.

(a) What is the probability that an arriving customer has to wait before being served? It is also called ‘blocking probability’.

(b) What is the probability that a server is busy at any particular point in time (denoted by ρ)? It is the proportion of time that a server actually spends with the customer i.e. the fraction of time a server is busy.

(c) What is the probability of ‘n’ customers being in the queuing system when it is in steady state condition? It is denoted by Pn , n = 0,1,2,3….

(d) What is the probability of service denial when an arriving customer cannot enter the system because the queue is full?

4. Cost related questions.

(a) What is the average cost needed to operate the system per unit of time?

(b) How many servers are needed to achieve cost effectiveness?

NOTATIONS

1. n = no. of customers in the system (waiting and in service).

2. Pn = Probability of ‘n’ customers in the system.

3. λ = Average (or expected) customer arrival rate or average arrivals per unit of time in the queuing system.

4. μ = Average (or expected) service rate or average number of customers served per unit time.

5. λ/μ = ρ = Average service completion time (1/μ)/Average inter arrival time (1/λ) = Traffic utilization or server utilization factor (the expected fraction of time for which the server is busy)

6. s or c = number of service channels (service facilities or servers).

7. N = finite number of customers allowed in the system or finite population.

8. Ls = Average (expected) no. of customers in the system (waiting and in service).

9. Lq = Average (expected) no. of customers in the queue (queue length).

10. Lb = Average (expected) length of non-empty queue.

11. Ws = Average (expected) waiting time in the system (waiting and in service).

12. Wq = Average (expected) waiting time in the queue.

Formulae for Poisson-exponential, single server model with infinite population.

Assumptions:

a) Arrivals follow Poisson distribution with mean arrival rate,

say λ.

b) The service time has exponential distribution with service

rate, say μ.

c) Arrivals are from infinite population.

d) Customers are served on FCFS basis.

e) There is only a single service station.

f) The system is in steady state of operation.

Formulae for Poisson-exponential, single server model with infinite population.

Under these conditions

1. Probability that the system is busy, ρ=λ/μ

2. Probability that the system is idle, P(0)=1-ρ

3. Probability that there are ‘n’ customers in the system (i.e. exactly ‘n’ customers), P(n)= ρn(1-ρ)

4. Probability that there are more than ‘n’ customers in the system, P(>n)=ρn+1

5. Expected number of customers in the system, Ls=ρ/(1-ρ) or λ/(μ-λ)

6. Expected number of customers in the queue, Lq=ρ2/(1-ρ) or λ2/μ(μ-λ)

7. Expected length of non-empty queues i.e. at least one customer, Lq

’=1/(1-ρ) or μ/(μ- λ)

8. Expected waiting time of a customer in the system, Ws=1/(μ- λ)

9. Expected waiting time of a customer in the queue, Wq=ρ/(μ-λ) or λ/μ(μ-λ)

10.Probability that a customer spends more that ‘t’ units of time in the system, Ws(t)=e-t/Ws

11.Probability that a customer spends more than ‘t’ units of time in the queue, Wq(t)= ρ.e-t/Ws

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Queueing Theory Ppt