Waiting Line Models

(Queuing Theory)

© 2012 Lew Hofmann

Lines in Operations Management

• Assembly lines

• Production lines

• Trucks waiting to unload or load

• Workers waiting for parts

• Customers waiting for products

• Broken equipment waiting to be fixed

• Customers waiting for service

© 2012 Lew Hofmann

Costs The cost of waiting

Losing customers because of long lines• Reneging: Customers get tired of waiting and leave

• Balking: Customers see a long line and don’t get in line.

Unusable (idle) equipment awaiting repairs• EG: Broken assembly line machinery.

The cost of service Paying people to provide service to customers Customers can be people, machines, or other objects needing service.

© 2012 Lew Hofmann

Cost of Providing Service

• Paying repairmen to fix broken machines

• Paying dock workers to load and unload trucks

• Paying customer-service people

• Using more production people to speed up the line

• Leasing of service equipment and facilities

• Paying checkout cashiers

© 2012 Lew Hofmann

Queuing System Costs

Number of Servers

Costs

Cost of Servicegoes up as you pay for more servers.

Costs of Waitinggoes down as service improves.

Total Cost

Optimal # of servers

Note that the lowest cost system requires some customer waiting.

Fewer servers often means longer waiting for customers. Many servers

means little or no waiting, but higher service costs.

© 2012 Lew Hofmann

Queuing theory in quantitative technique is useful for determining the optimum number of service facilities

Managerial application of queuing theory

1. Aircraft at landing and take-off from busy airport.

2. Routing sales persons 3. Jobs in production control

© 2012 Lew Hofmann

What Queuing Models Tell Us.

• Average time in line for a customer.

• Average number of customers in line.

• Average time in the system for a customer.

• Average number of customers in the system at any time.

• Probability of n number of customers in the system at any given time.

• NOTE: “In The System” includes customers who are in line plus the customers being served.

© 2012 Lew Hofmann

ARRIVAL SYSTEM(How customers arrive)

QUEUE(The nature of the waiting line or lines of customers)

The Waiting Line System

SERVICE FACILITY(How customers progress through the service facility)

© 2012 Lew Hofmann

Waiting Line Models

Served customers

Arrival System

Service System

Waiting line (Queue)

Priority rule

Service facilities

The sequence in which customers are admitted into the service facility.

Population

© 2012 Lew Hofmann

Arrival System

• Arrival Populations are either…

• Limited (EG: Counters for handicapped persons.)

• Unlimited (EG: cars arriving at a toll booth)

• Arrival Patterns are either…

• Random (Each arrival is independent e.g.in Bank)

• Scheduled (EG: Doctor’s office visits)

• Behavior of the Arrivals

• Balking (Seeing a long line and avoiding it.)

• Reneging (Get tired of waiting and leave the line)

• Jockeying (Switching lines)

© 2012 Lew Hofmann

The Queue

• Queue Length is either..

• Unlimited (EG: cars in line at a toll booth)

• Limited (Finite) EG: # of e-mail messages allowed.

• Queue Discipline (order of service)

• FIFO (First-In, First-Out)

• LIFO (Last-In, First-Out)

• Priority

© 2012 Lew Hofmann

The Service Facility

• Channels• How many paths (ways to get through the

system) are there after getting in line?• In Bank Customer can use

• Channel I: ATM machine to withdraw cash

• Channel II:May stand in queue to get it from cashier.

© 2012 Lew Hofmann

Single-channel, Single-phaseOne way through the system

and one stop for service

Service Facility

© 2012 Lew Hofmann

Multi-channel, Single-phase

Service Facility

Service Facility

Once in line, you have at least two choices of how to get through the system, but only one stop.

© 2012 Lew Hofmann

Multiple v.s. Single Customer Queue Configuration

1. The service provided can be differentiated Ex. Supermarket express lanes

2. Labor specialization possible

3. Customer has more flexibility

4. Balking behavior may be deterredSeveral medium-length

lines are less intimidating than one very long line

1. Guarantees fairness FIFO applied to all arrivals

2. No customer anxiety regarding choice of queue

3. Avoids “cutting in” problems

4. The most efficient set up for minimizing time in the queue

5. Jockeying (line switching) is avoided

Multiple Line Advantages Single Line Advantages

© 2012 Lew Hofmann

Components of a Basic Queuing Process The Service Mechanism

Can involve one or several service facilities with one or several parallel service channels (servers) - Specification is required

The service provided by a server is characterized by its service time• Specification is required and typically involves data

gathering and statistical analysis.• Most analytical queuing models are based on the

assumption of exponentially distributed service times, with some generalizations.

The queue discipline Specifies the order by which jobs in the queue are being

served. Most commonly used principle is FIFO. Other rules are, for example, LIFO, SPT, EDD… Can entail prioritization based on customer type.

© 2012 Lew Hofmann

n = total number of customers in the system at time ‘t’

l = average number of customers arriving per unit time

µ = average number of customers being serviced per unit time

C = number of parallel service channels

Ls= average number of customers in the system = /(µ - )

Lq = average number of customers waiting in the queue = x / µ(µ - )

NOTATIONS : (M/M/1) [ follows Poisson Distribution and µ follows Exponential Distribution ]

© 2012 Lew Hofmann

NOTATIONS :

Ws = average number of time of customer in the system in both waiting and in service. = 1/ (µ - )

Wq = average waiting time of customer in a queue = / µ(µ - )

Pn(t) = Prob. That there are n customers in the system at any time t both waiting and in service= ( 1- / µ) ( / µ)^ n

Utilization factor = [Average service completion time(1/ µ)]/ [Average inter-arrival time (1/ )] P[system is busy] = = / µ = Expected number of customers in the systemP[ system is idle] = 1- P[ system is busy] = 1 -

© 2012 Lew Hofmann