Waiting_Line_Management.pdf

7/27/2019 Waiting_Line_Management.pdf

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Waiting Line Management

Ravindra S. Gokhale

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Examples of Waiting Lines

Customerswaiting at the single window counter of a bank

Customers waiting at a restaurant for food to be served (after placing the

order)

Children waiting at an amusement park for a particular ride

Documents waiting for their turn to be processed

Data network where packets arrive, wait in various queues, receive service

at various points, and exit after some time (computer networking)

A telephone call waiting to get through in a busy telecommunication

network

Jobs waiting for their turn to be processed on a CNC machine

Machines waiting for repairs

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Early Studies on Waiting Line Problems

A. K. Erlang, a Danish engineer who worked for the Copenhagen

Telephone Exchange, published the first paper on queuing theory in 1909

David G. Kendall introduced an A/B/C queuing notation in 1953

It was later on extended as A/B/C/K/N/D notation

Leonard Kleinrock worked on queuing theory in the early 1960s

His work is used in modern packet switching networks

John Little published the proof of Littles Law in 1961

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Structure of the Waiting Line Problem

Arrival process

It is the input process

The distribution that determines how the tasks arrives in the system

May be deterministic (example: prior appointment) or probabilistic

Most common distribution for modeling arrivals: Poisson

Arrivals are known as customers

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Structure of the Waiting Line Problem (cont)

Service process

Describes the processing of tasks

The distribution that determines how the tasks leave the system

May be deterministic (example: computerized controlled process) or

probabilistic

Most common distribution for modeling services: Exponential, Erlang

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Number of servers

Total number of parallel servers/stations available to process the task

Single server, multiple servers

In the basic models, the servers are assumed to be of equal capability

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Capacity of the queue

Number of arrivals that the queue can hold

Assumed to be infinite for the basic models

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Calling population

Population of potential arrivals

Infinite (example: shoppers arriving at a shopping mall) or finite (example:

machines arriving for maintenance)

For finite (and small) population, the effective arrival rate decreases aftereach arrival and increases after each service completion

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(Queue) Discipline

FCFS First come first served

SPT Shortest processing time first

EDD Earliest due date first

SIRO Service in random order

Pre-emptive Example: In healthcare systems, a higher priority arrival can

interrupt some established discipline or even some ongoing service, due to

medical emergency

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Nomenclature

Kendalls notation

A/B/C/K/N/D

A = Arrival distribution

B = Service distribution

C = Number of servers

K = Capacity of the system (considered infinite if not specified)

N = Calling population (considered infinite if not specified)

D = Discipline of the queue (considered as FCFS if not specified)

Examples

M/M/1///FCFS (denoted just as: M/M/1) M/D/1///FCFS (denoted just as: M/D/1)

M/M/s///FCFS (denoted just as: M/M/s)

M/M/s//N/FCFS

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Customer Behavior

Patient customer

Joins the queue and waits till his/her turn for servicing

Impatient customer

Balking Customer arrives, but decides not to join the queue due to longer

queue length

Reneging After being in the queue for some time, the customer leaves the

system, without taking the service

Jockeying A customer switches from one queue to another anticipating

faster service

*In this course we will assume only patient customer behavior

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A queue forms not just because the arrival rate is greater

than the service rate*, but because the nature of arrivals

(and service times) is probabilistic.

*The systems where arrival rate is greater than the service rate (and the

calling population is infinite) are unstable systems and cannot be

sustained after some point of time.

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Different Types of Systems

Based on channels and phases

Channel: Number of options available for the customer

Phase: Number of stages in the service

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Number of phases

Numberofchannels

Multiple

Single

Multiple

Single

One channel sufficient andservice offered in a singlestage

Single server hair cutting shop

Single server cobbler shop

One channel sufficient, butservice can best be offered inmultiple stages

Small or medium groceryshop with different stages placing order, getting goods,paying cash

Multiple channels necessary

and but service requiressingle stage

Single window counter in abank with multiple tellers

Toll plaza

Multiple channels necessary

and service requires differentstages

Automobile assembly line withmultiple lines and multipleworkstations

Large petrol filling station


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Performance Measures for Analysis (Quantitative)

Average number of customers in the queue (Lq)

Waiting to be served

Average number of customers in the system (Ls)

Includes those waiting to be served as well as those being served

Average waiting time in the queue (Wq)

Waiting time before commencement of service

Average waiting time in the system (Ws)

Includes the total time i.e. from joining the queue till completion of service

Service facility utilization ()

Signifies how busy is the service channel(s)

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Littles Law

The long-term average number of customers in a stable system Ls is equal to

the long-term average arrival rate, , multiplied by the long-term average

time a customer spends in the system, Ws

Ls = Ws

Holds good for a wide variety of systems

Does not require knowledge about distribution of arrivals and service times

Littles Law cannot be applied to systems with finite calling population

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Littles Law

Practical applications:

Manufacturing If one measures the average time a product spends in the

system i.e. Ws and the rate of arrival of products i.e. , one can compute the

Work in Process (WIP) Inventory i.e. Ls

Services If one counts the average number of customers in a system Ls and

the rate of arrival of customers , then one can compute the average waiting

time spent by a customer in the system Ws

Littles Law cannot be applied to systems with finite calling population

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Single Server Model : M/M/1

Arrival rate follows a Poisson distribution

That means, time between the arrivals follows an Exponential distribution

Arrival rate is denoted bycustomers per unit time

Service times follow an Exponential distribution

That means, service rate follows a Poisson distribution

Service rate is denoted bycustomers per unit time

For the system to be stable, >

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Single Server Model : M/M/1 (cont)

Server utilization: = /

Probability ofn customers in the system: Pn = (1 ) n

Probability that there are no customers in the system:P0 = (1 )

(This is, the probability that the server is idle)

Average number of customers in the queue: Lq = 2/[ ()]

Average waiting time in the queue: Wq = Lq /

Average number of customers in the system: Ls = Lq +( / )

Average waiting time in the system: Ws = Ls / = Wq +(1 / )

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Single Server Model : M/M/1 (cont)

Customer arrival at a single server post office counter is Poisson with an

average rate of 6 customers per hour. The service times are exponential.

The server is idle for 20% of the time, on an average.

a. Determine the average service time, average length of the queue,

average waiting time before being served, average time spent in thesystem and average number of people in the system.

b. What is the probability that there are more than 4 customers in the

system?

c. What is the desired average service time if the management wants to

ensure that only for 10% of the time, the number of customers in the

system exceed four?

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Numerical example


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Single Server Model with Deterministic ServiceTimes : M/D/1




Service times are deterministic

Applicable in situations where the service process is automated


For the system to be stable, >

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Single Server Model with Deterministic ServiceTimes : M/D/1 (cont)

Server utilization: = /

Probability that there are no customers in the queue:P0 = (1 )

(This is, the probability that the server is idle)

Average number of customers in the queue: Lq

= 2/[2 ()]

Average waiting time in the queue: Wq = Lq /

Average number of customers in the system: Ls = Lq +( / )

Average waiting time in the system: Ws = Ls / = Wq +(1 / )

Numerical examples

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Multiple Server Model : M/M/s







Number of servers = s

For the system to be stable, (s ) >

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Multiple Server Model : M/M/s (cont)

Server utilization: = / (s )

Probability that an arrival has to wait:

(This is, the probability that all the servers are busy)

where P0 = Probability that there are no customers in the system, that is, probability

that all servers are idle

Average number of customers in the system:

Average waiting time in the system: Ws = Ls /

Average number of customers in the queue: Lq = Ls -( / )

Average waiting time in the queue: Wq = Lq / = Ws -(1 / )

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Customer arrival at a bank is Poisson with an average rate of 24 customers

per hour. There are three counters in the bank and each service counter

has exponential service time with an average of 6 minutes per customer.

With this data, there is a 6% probability that all the servers are idle.

a. Determine the service facility utilization, average length of the queue,

average waiting time before being served, average time spent in the

system and average number of people in the system.

b. What is the probability that all the servers are busy?

c. If the efficiency of the service facility is improved and the average

service time is brought down to 5 minutes per customer, can the

management have only two counters for the bank?

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Numerical example 1


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Maintenance department in a large manufacturing organization has the

responsibility to look after the maintenance of all the machines in the

organization. Assume that all the machines work round the clock. The

maintenance department functions as independent crews, which are

assumed to be equally capable. Each crew has three personnel involved.

Currently there are two such crews. The time between failure of machinesfollows an exponential distribution, and on an average 2 machines fail

everyday. The repair time of a maintenance crew for one machine is

exponentially distributed with a mean of 20 hours. With this information,

there is a 0.09 probability that both the crews are idle. The average idle

time cost of a machine is Rs. 300 per hour. Each maintenance personnel is

paid Rs. 75 per hour. What is the average cost of each breakdown?

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Numerical example 2


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Single Server Model with Finite CallingPopulation: M/M/1//N/FCFS

Arrival rate follows a Poisson distribution That means, time between the arrivals follows an Exponential distribution





Calling population is finite = N customers

Littles Law cannot be applied in this case

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Single Server Model with Finite CallingPopulation: M/M/1//N/FCFS (cont)

Average number of customers in the queue:

where P0 = Probability that there are no customers in the system, that is,

the server is idle

Average number of customers in the system: Ls = Lq +(1 P0)

Average waiting time in the queue: Wq = Lq /[ (N - Ls)]

Average waiting time in the system: Ws = Wq +(1 / )

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Single Server Model with Finite CallingPopulation: M/M/1//N/FCFS (cont)

XYZ Manufacturing Company has 20 machines. Each machine operates an

average of 200 hours before breaking down. The maintenance staff

includes 5 workers and works together as a team. The average time to

repair is 3.6 hours. Breakdown rate is Poisson distributed and service time

is exponentially distributed. With this data, the probability that the

maintenance staff is idle turn out to be 0.65. Find the average number of

machines in the repair queue, average number of machines in the repair

system, average waiting time in the repair queue and average waiting time

in the repair system.

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Numerical example


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Behavioral and Other Considerations

Non-linear waiting costs and the psychology of waiting

Customers have some threshold for waiting

Nature of emergency

Social justice

Perception of inequality

Environment

How pleasant is the waiting environment

Service providers can modify the environment so that waiting may not

seemlong

Feedback and accurate information about expected waiting time

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Some Mitigation Techniques

Allow customers to serve themselves (at least partially)

Shorten service times by performing tasks in advance

Use two queues for each server

Reward customers for being patient and waiting

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