Waiting Line Management

Saurabh Chandra

Where the Time GoesIn a life time, the average

American will spend--

SIX MONTHSWaiting at stop lights

EIGHT MONTHSOpening junk mail

ONE YEARLooking for misplaced objects

TWO YEARS Unsuccessfully returning phone calls

FOUR YEARS Doing housework

FIVE YEARS Waiting in line

SIX YEARS Eating

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Waiting Realities & Perceptions • Inevitability of Waiting: Waiting results from variations in arrival

rates and service rates• Economics of Waiting: High utilization purchased at the price of

customer waiting. Make waiting productive (salad bar) or profitable (drinking bar).

• Skinner’s Law:The other line always moves faster.

• Jenkin’s Corollary:However, when you switch to another other line, the line you left moves faster.

Waiting Line Analysis• Waiting occurs in production (?) and service processes.• Time is a valuable resource and hence reduction of waiting time

desirable.• Waiting line exists as

Customers (people or things) arrive faster than they can be served.

Customers do not arrive at constant rate Service time is also not constant

5-4

5-5

Traditional Cost Relationships• as service improves, cost increases

5-6

Elements ofWaiting Line Analysis• Waiting line system

• consists of arrivals, servers, and waiting line structure• Queue

• a single waiting line; finite or infinite• Infinite queue• Finite queue

• can be of any length; length of a finite queue is limited

• Calling population• source of customers; infinite or finite

5-7

Essential Features of Queuing Systems

DepartureQueuediscipline

Arrival process

Queueconfiguration

Serviceprocess

Renege

Balk

Callingpopulation

No futureneed for service

Arrival Process

Static Dynamic

AppointmentsPriceAccept/Reject BalkingReneging

Randomarrivals withconstant rate

Random arrivalrate varying

with time

Facility-controlled

Customer-exercised

control

Arrival process

Distribution of Patient Inter-arrival Times at a Doctor’s clinic

0

10

20

30

40

1 3 5 7 9 11 13 15 17 19

Rela

tive

freq

uenc

y, %

Patient interarrival time, minutes

Poisson and Exponential Equivalence

Poisson distribution for number of arrivals per hour (top view)

One-hour

1 2 0 1 interval

Arrival Arrivals Arrivals Arrival

62 min.40 min.

123 min.

Exponential distribution of time between arrivals in minutes (bottom view)

Queue Configurations Multiple Queue Single queue

Take a Number Enter

3 4

8

2

6 10

1211

5

79

Are these two systems same?

Queue Discipline

Queuediscipline

Static(FCFS rule) Dynamic

selectionbased on status

of queue

Selection basedon individual

customerattributes

Number of customers

waitingRound robin Priority Preemptive

Processing timeof customers

(SPT rule)

• What is meant by Queue Discipline?• What’s the most common one?

• Examples of systems with different QDs?

Outpatient Service Process Distributions

0

5

10

15

1 11 21 31 41

Relativ

e freq

uenc

y. %

Minutes

0

5

10

15

1 11 21 31 41Relativ

e freq

uenc

y,

%

Minutes

0

5

10

15

1 11 21 31 41Relativ

e freq

uenc

y,

%

Minutes

Elements ofWaiting Line Analysis• Queue discipline

• order in which customers are served• First come first served• Last in first out• Random• Others

Elements of Waiting Line Analysis

• Basic Waiting Line Structures Single Channel Single phase (single server) Multiple Channel Single phase (Multiple server) Single Channel Multiple phase Multiple Channel Multiple phase

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Configurations

DeparturesAfter Service

Single-Server, Single-Phase System

Queue

Arrivals Service Facility

Single-Server, Multiphase System

Arrivals Departuresafter Service

Phase 1 Service Facility

Phase 2 Service Facility

Queue

Configurations

Multi-Server, Single-Phase System

Arrivals

QueueDeparturesService

Facility1

Service Facility

2

Service Facility

3

after

Service

Configurations

Multi-Server, Multiphase System

Arrivals

QueueDeparturesafter service

Type 2 Service Facility

1

Type 2 Service Facility

2

Type 1 Service Facility

1

Type 1 Service Facility

2

Elements ofWaiting Line Analysis

• Channels• number of

parallel servers for servicing customers

• Phases• number of

servers in sequence a customer must go through

Arrival Characteristics

• Size of the arrival population• Infinite or finite

• Arrival distribution• Arrival rate• Average arrival time• Poisson distribution

• Behavior• Patient, • Balking: leaving on seeing a line • Reneging: leaving the line before service

Poisson Distribution

where

X = number of arrivals per unit of time (e.g., hour)

P(X) = probability exactly X arrivalsl = average arrival rate (i.e., average

number of arrival per unit of time)e = 2.7183 (known as the exponential

constant)

Poisson Distribution

Pro

babi

lity

l = 2 Distribution l = 4 DistributionX

0.25

0.20

0.15

0.10

0.05

X0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 11

Figure 9.2

Queue Characteristics

• Length• Finite (limited) or infinite (unlimited)

• Discipline• FIFO common• Other ways to prioritize arrivals

Service Characteristics• Configuration

• Servers (channels) and phases (service stops)• Single-server, multiple-server• Single phase, multiphase system

• Service Distribution• Constant or random• Exponential distribution• Service rate, service time

Exponential Distribution

where

t = service timeP(t) = probability that service time will be

greater than tm = average service rate (i.e., average

number of customers served per unit of time)e = 2.7183 (known as the exponential

constant)

Exponential DistributionProbability That Service Time ≥ t = e–mt for t ≥ 0

m = Average Service Rate

Average Service Rate = 1 customer per hour

Average service Rate = 3 Customers per Hour Average Service Time = 20 Minutes (or 1/3 Hours)

per Customer

| | | | | | | | | | | | |0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

1.0 –

0.9 –

0.8 –

0.7 –

0.6 –

0.5 –

0.4 –

0.3 –

0.2 –

0.1 –

0.0 –

Pro

babi

lity

That

Ser

vice

Tim

e ≥

t

Time t in Hours

Measuring Queue Performance

• r = utilization factor of the system (i.e., probability all servers are busy)

• Lq = average length (i.e., the number ofcustomers) of the queue

• L = average number of customers in thesystem (i.e., the number in the queueplus the number being served)

• Wq = average time that each customer spends in the queue

Measuring Queue Performance

• W = average time that each customerspends in the system (i.e., the timespent waiting plus the time spent being served)

• P0 = probability that there are no customers in the system (i.e., the probability that the service facility will be idle)

• Pn = probability that there are exactly n customers in the system

Kendall’s NotationA / B / s / Lmax / POPsize

where

A = the arrival probability distribution. Typically choices are M (Markovian) for a Poisson distribution, D for a constant or deterministic distribution, or G for a general distribution with a known mean variance.

B = the service time probability distribution. Typical choices are M for an exponential distribution, D for a constant or deterministic distribution, or G for a general distribution with a known mean and variance.

s = number of servers.

Queuing Models StudiedNAME(KENDALL # OF TIME POPLN.NOTATION) EXAMPLE SERVERS PATTERN SIZESimple system Information Single Exponential Unlimited(M/M/1) counter at

department storeMultiple-server Airline Multiple Exponential Unlimited(M/M/s) ticket

counterConstant service Automated Single Constant Unlimited(M/D/1) car washGeneral service Auto repair Single General Unlimited(M/G/1 shopLimited Shop with Multiple Exponential Limitedpopulation exactly ten(M/M/s/∞/N) machines that

might break

All models are single phase with a Poisson arrival pattern and a FIFO queue discipline

Table 9.2

Queuing Models Studied1. Arrivals follow the Poisson probability

distribution2. FIFO queue discipline3. A single-phase service facility4. Infinite, or unlimited, queue length. That is,

the fourth symbol in Kendall’s notation is ∞5. Service systems that operate under steady,

ongoing conditions. This means that both arrival rates and service rates remain stable during the analysis.

M/M/1 Model• Assumptions

1. Arrivals are served on a FIFO basis.2. Every arrival waits to be served, regardless of the length

of the line; no balking or reneging.3. Arrivals are independent, the average number of arrivals

(the arrival rate) is constant.4. Arrivals are described by a Poisson probability

distribution, infinite or very large population.5. Service times vary from one customer to the next, are

independent of each other, with a known average rate.6. Service times occur according to the exponential

probability distribution.7. The average service rate is greater than the average

arrival rate; that is, m > l.

Operating Characteristicsl = average number of arrivals per time period (e.g., per hour )m = average number of people or items served per time period

1. Average server utilization in the system:

2. Average number of customers or units waiting in line for service:

3. Average number of customers or units in the system:

Operating Characteristics4. Average time a customer or unit spends waiting in line for service:

5. Average time a customer or unit spends in the system:

6. Probability that there are zero customers or units in the system:

7. Probability that there are n customers or units in the system:

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Advanced Single-Server Models• Constant service times

• occur most often when automated equipment or machinery performs service

• Finite queue lengths• occur when there is a physical limitation to length of waiting line

• Finite calling population• number of “customers” that can arrive is limited

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Advanced Single-ServerModels (cont.)

M/M/S Model• Same assumptions apply• More than 1 server

l = average number of arrivals per time period (e.g., per hour )m = average number of customers served per time per servers = number of servers

• single waiting line and service facility with several independent servers in parallel

• same assumptions as single-server model• sμ > λ

• s = number of servers• servers must be able to serve customers faster than they arrive

Operating Characteristics1. Average server utilization in the system:

2. Probability that there are zero customers or units in the system:

3. Average number of customers or units waiting in line for service:

Operating Characteristics4. Average number of customers or units in the system:

5. Average time a customer or unit spends waiting in line for service:

6. Average time a customer or unit spends in the system:

7. Probability that there are n customers or units in the system:

M/D/1 Model• Service rate is constant• Waiting times and number of customers/units

always less than M/M/s system

l = average number of arrivals per time period (e.g., per hour )m = constant number of people or items served per time period

Operating Characteristics1. Average server utilization in the system:

2. Average number of customers or units waiting in line for service:

3. Average number of customers or units in the system:

Operating Characteristics4. Average time a customer or unit spends waiting in line for service:

5. Average time a customer or unit spends in the system:

6. Probability that there are zero customers or units in the system:

M/G/1 Model• Service time follows a general distribution

l = average number of arrivals per time period (e.g., per hour )m = average number of people or items served per time periods = standard deviation of service time

Operating Characteristics1. Average server utilization in the system:

2. Average number of customers or units waiting in line for service:

3. Average number of customers or units in the system:

Operating Characteristics4. Average time a customer or unit spends waiting in line for service:

5. Average time a customer or unit spends in the system:

6. Probability that there are zero customers or units in the system:

M/M/S/∞/N Model• Dependent relationship between queue length and

arrival rate• Assumptions

1. There are s servers with identical service time distributions.

2. The population of units seeking service is finite, of size N.

3. The arrival distribution of each customer in the population follows a Poisson distribution, with an average rate of l.

4. Service times are exponentially distributed, with an average rate of m.

5. Both l and m are specified for the same time period.6. Customers are served on a first-come, first-served basis.

Operating Characteristicsl = average number of arrivals per time period (e.g., per hour )m = average number of people or items served per time periods = number of serversN = size of population

1. Probability that there are zero customers or units in the system:

Operating Characteristics2. Probability that there are exactly n customers in the system:

3. Average number of customers or units in line, waiting for service:

Operating Characteristics4. Average number of customers or units in the system:

5. Average time a customer or unit spends in the queue waiting:

6. Average time a customer or unit spends in the system:

More Complex Systems

• Variations may be present• More complex models have been developed• May require computer simulation

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Psychology of Waiting• That Old Empty Feeling: Unoccupied time goes slowly• A Foot in the Door: Pre-service waits seem longer that

in-service waits• The Light at the End of the Tunnel: Reduce anxiety

with attention• Excuse Me, But I Was First: Social justice with FCFS

queue discipline• They Also Serve, Who Sit and Wait: Avoids idle service

capacity

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Psychology of Waiting• Waiting rooms

• magazines and newspapers• Televisions

• Use of• Mirrors

• Supermarkets• magazines• “impulse purchases”

• Disney• costumed characters• mobile vendors• accurate wait times• special passes

• Preferential treatment• Grocery stores: express lanes for customers with few purchases• Airlines/Car rental agencies: special cards available to frequent-users or for

an additional fee• Phone retailers: route calls to more or less experienced salespeople based on

customer’s sales history• Critical service providers

• services of police department, fire department, etc.• waiting is unacceptable; cost is not important

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Psychology of Waiting (cont.)