Operations Management

Program: PGPBA Course Code: IBS 537Semester: II Sessions : 33Class of: 2008 Credits : 3

14. Waiting Line Management

Capacity Vs Waiting Line

CostCost

Service Facility CapacityService Facility Capacity

Cost of capacity

Waiting costs

Total costsMinimum

cost

Optimal capacity

Components of the Queuing System

ArrivalsArrivals

ServersServers

QueueQueue

Service SystemService System

DeparturesDepartures

Arrival Characteristics

Input Source(Population)

Size BehaviorArrival Pattern

FiniteInfinite Random Non-Random Patient Impatient

Balk RenegePoisson Other

Waiting Line Characteristics

Waiting Line

Length QueueDiscipline

LimitedUnlimited FIFO(FCFS) Random Priority

Service Characteristics

ServiceFacility

Configuration

Multi-Channel

SingleChannel

SinglePhase

Factors Affecting the Queuing Systemn Queue length capacityn Number of queuesn Queue disciplinen Service time distributionn Line structure

q Single channel, single phaseq Single channel, multi-phaseq Multi-channel, single phaseq Multi-channel, multi-phaseq Mixed

n Customer departure rate

Suggestions for Managing Queuesn Determine an acceptable waiting timen Divert your customer attentionn Inform customers of expectationsn Keep non-service employees out of sightn Segment customersn Friendly serversn Encourage arrivals during slack periodsn Take long-term perspective toward reducing queues

Common Queuing Models

These models share the following characteristics:• Single phase• Poisson arrivals• FCFS• Infinite queue length

Model LayoutSourcePopulation Service Pattern

1 Single channel Infinite Exponential

2 Single channel Infinite Constant

3 Multi-channel Infinite Exponential

4 Single or Multi Finite Exponential

Single Channel – Single Phase

ArrivalsServed units

Service facility

Queue

Service system

Dock

Waiting ship lineShips at sea

Ship unloading system Empty ships

Single Channel, Multi-phase

Cars& food

ArrivalsServed units

Service facility

Queue

Service system

Pick-upWaiting cars

Cars in area

McDonald’s drive-through

Pay

Service facility

Multi Channel, Single Phase

Arrivals

Served units

Service facilityQueue

Service system

Service facility

Example: Bank customers wait in single line for one of several tellers.

Multi-channel, Multi-phase

Service facility

Arrivals

Served units

Service facilityQueue

Service system

Service facility

Example: At a laundromat, customers use one of several washers, then one of several dryers.

Service facility

Queuing Notation = Arrival rateλ

s

Average # in system

(including being served)L =

Average time waiting in lineqW =

= Service rateµ

=sWAverage total time in system(including time to be served)

1 Average service timeµ

=

# of units in system n =

1 Average time between arrivals λ

=

# of identical service channelsS =

Ratio of total arrival rate to = =

single server sevice rateλ

ρµ

Prob. of exactly units in systemnP n=

Average # waiting in lineqL =

Probability of waiting in linewP =

Equations for Solving Queuing Models

Model 1 Examplen Assume a drive-up window at a fast food restaurant.

Customers arrive at the rate of 25 per hour with Poisson arrivals. The employee can serve one customer every two minutes on average, with exponential service rates.

n Determine:a) the average utilization of the employeeb) the average number of customers in linec) the average number of customers in the systemd) the average waiting time in linee) the average waiting time in the systemf) the probability that exactly two cars will be in the system

Model 2 Examplen An automated pizza vending machine heats and dispenses a

slice of pizza in 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution.

n Determine:a) the average number of customers in lineb) the average total waiting time in the system

Model #3 – Expected Number in Line (Lq) Tables

Note: Added this slide 02 Oct 2006.

Lq

Model 3 Examplen Recall the Model 1 example:

q Drive-up window at a fast food restaurant.q Customers arrive at the rate of 25 per hour.q The employee can serve one customer every two minutes.q Assume Poisson arrival and exponential service rates.

n If an identical window (and an identically trained server) were added, determine the effect on:a) the average number of cars in the systemb) the total time customers wait before being served

More Queuing Notation (for Model 4)

Probability that an arrival

must wait in lineD =

Service factor, or proportionof service time required per customer

X =

( )Population source, less those

in queuing system -

JN n

=

Average # of units in lineL =

Efficiency factor, measures the effect of waiting in line

F =

Average # of units being served H =

# of service channelsS =

Average waiting time in lineW =

Average time betweencustomer service requirements

U =

Average service timeT =

Prob. exactly units in systemnP n=

# of units in population sourceN =

Average # of units in system(including being served)

n =

Model #4 – Efficiency Factor (F) Tables

Note: Added this slide 02 Oct 2006.

Model 4 Examplen The copy center of an electronics firm has four copy machines

that are all serviced by a single technician. Every two hours, on average, the machines require adjustment. The technician spends an average of 10 minutes per machine when adjustment is required. Assume Poisson arrivals and exponential service.

n Determine:q the average number of machines that are down

Queuing System Approximationsn Detailed knowledge of arrival and service time distributions

don’t strictly need to be done.q a quick approximation exists that can provide reasonably

accurate analysis of the queuing modelq inter-arrival and service time distributions assumed to be generalq average performance measures (waiting time in queue, number

in queue, etc) can be very well approximated with only the mean and variance of the distributions (exact shape is unimportant)

n First, we need to define:

aa

a

SCX

= ss

s

SCX

=

Standard deviation of Xcoefficient of variationMean of XxC = =

Queuing System Approximations (2)n Inputs:

q # servers, Sq Arrival rate, λq Service rate, µq Coefficients of variance,

n Calculate:q Ratio of inter-arrival and service rates,

q Mean # in queue and in system,

q Mean waiting time in queue and system,

2( 1) 2 2

1 2

Sa s

qC CL ρ

ρ

+ += ×

−

s qL L Sρ= +

Sλ

ρµ

=

2 2,a sC C

qq

LW

λ= s

sLWλ

=

Queuing System Approximations Examplen Consider a make-to-order manufacturing process consisting

of a single stage with five machines. Processing times have a mean of 5.4 days and standard deviation of 4 days. Management has collected data on customer orders, and verified that the time between orders has a mean of 1.2 days and variance of 0.72 days.

n Determine:q the expected machine utilizationq the expected number of orders waiting for processingq the average time that an order waits before being worked on

Review of the Queuing Modelsn Model #1

q single channel, infinite source population, exponential service rate

q e.g., main reception, customer service deskn Model #2

q single channel, infinite source population, constant service rateq e.g., vending machine, automatic car wash, automated services

n Model #3q multi-channel, infinite source population, exponential service rateq e.g., bank tellers

n Model #4q Single or multi-channel, finite source population, exponential

service rateq e.g., internal service/maintenance