Service Systems &

Queuing

Chapter 12SOPS 370

Nature of Services

• 1. • 2.

– A. • 3.

• 4. • 5. • 6. • 7.

Service System Design Matrix

Mail contact

Face -to-faceloose specs

Face -to-facetight specs

PhoneContact

Face -to-facetotal

customization

(Buffered System) None

(Permeable System) Some

(Reactive System)Extensive

(high)

(low)High

Low

Degree of customer/server contact

Internet & on-site

technology

SalesOpportunity?

(ProductionEfficiency?)

Designs for On-Site Service

• 1.

– Ex:• 2.

– Ex. • 3.

– Ex.

Disney World

• 1.

• 2.

• 3.

• 1. • 2. • 3. • 4.

Implications of Waiting Lines

Elements of Waiting Lines• 1.• 2.

– A.

– B.

• 3.

• 4.

Customer Population Characteristics

• 1. – A.

• 2. – A.

• 3. – A.

• 4. Jockeying– A.

Service System

• 1. The service system is defined by:– A. – B. – C. – D. – E.

Number of Lines

• 1. Waiting lines systems can have single or multiple queues.– A.

– B.

Servers• 1.

• 2.– A.

– B. Example of a multi-phase, multi-server system:

C C C CC DepartArrivals1

2

3 6

5

4

Phase 1 Phase 2

Example Queuing Systems

Arrival & Service Patterns• Arrival rate:

– 1. The average number of customers arriving per time period

– 2. Modeled using the Poisson distribution– 3. Arrival rate usually denoted by lambda ()– 4. Example: =50 customers/hour; 1/=0.02 hours

between customer arrivals (1.2 minutes between customers)

Arrival & Service Patterns (Continued)

• Service rate:– 1. The average number of customers that can be served during

the period of time– 2. Service times are usually modeled using the exponential

distribution– 3. Service rate usually denoted by mu (µ)– 4. Example: µ=70 customers/hour; 1/µ=0.014 hours per

customer (0.857 minutes per customer).• Even if the service rate is larger than the arrival rate,

waiting lines form!– 1. Reason is the variation in specific customer arrival and

service times.

Waiting Line Priority Rules• 1. First come, first served• 2. Best customers first (reward loyalty)• 3. Highest profit customers first • 4. Quickest service requirements first• 5. Largest service requirements first• 6. Earliest reservation first• 7. Emergencies first

Waiting Line Performance Measures

• Lq = The average number of customers waiting in queue

• L = The average number of customers in the system• Wq = The average waiting time in queue• W = The average time in the system• r = The system utilization rate (% of time servers are

busy)

Single-Server Waiting Line• Assumptions

– 1. Customers are patient (no balking, reneging, or jockeying) – 2. Arrivals follow a Poisson distribution with a mean arrival

rate of . This means that the time between successive customer arrivals follows an exponential distribution with an average of 1/

– 3. The service rate is described by a Poisson distribution with a mean service rate of µ. This means that the service time for one customer follows an exponential distribution with an average of 1/µ

– 4. The waiting line priority rule is first-come, first-served– 5. Infinite population

Formulas: Single-Server Case

lambda mean arrival ratemu mean service rate

average system utilization

Note: for system stability. If this is not the case,an infinitly long line will eventually form.

r

Formulas: Single-Server Case con’t

average number of customers in system

average number of customers in line

1 average time in system including service

average time spent waiting

1 probability of customers in the system

at a

q

q

nn

L

L L

W

W W

P n

r

r

r r

given point in time

State Univ Computer Lab• A help desk in the computer lab serves students on a

first-come, first served basis. On average, 15 students need help every hour. The help desk can serve an average of 20 students per hour.

• Based on this description, we know:– 1. µ = 20 students/hour (average service time is 3 minutes)– 2. = 15 students/hour (average time between student

arrivals is 4 minutes)

Average Utilization

15 0.75 75%20

orr

Average Number of Studentsin the System, and in Line

studentsL 31520

15

0.75 3 2.25qL L studentsr

Average Time in the System & in Line

minutes12

hours2.01520

11

or

W

0.75 0.2 0.15 hours

9 minutesqW W

or

r

Probability of nStudents in the Line

00

11

2 22

3 33

4 44

1 1 0.75 1 0.25

1 1 0.75 0.75 0.188

1 1 0.75 0.75 0.141

1 1 0.75 0.75 0.105

1 1 0.75 0.75 0.079

P

P

P

P

P

r r

r r

r r

r r

r r

Single Server: Probability of n Students in the System

Probability of Number in System

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

0.30000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number in System

Prob

abili

ty

Multiple Server Case

• Assumptions– 1. Same as Single-Server,

except here we have multiple, parallel servers

– 2. Single Line– 3. When server finishes

with customer, first person in line goes to the idle server

– 4. All servers are identical

Multiple Server Formulaslambda mean arrival ratemu mean service rate for server number of parallel, identical servers

average system utilization

Note: for system stability. If this is not the case,an infinitly l

ones

ss

r

ong line will eventually form.

Multiple Server Formulas con’t

11

00

0

0

/ / 1 probability of zero! ! 1

customers in the system at a given point in time

/ for

! probability of customers/

for !

in the s

n ss

n

n

n n

n s

Pn s

P n snP n

P n ss s

r

ystem at a given point in time

Multiple Server Formulas (Continued)

02

/ average number of customers in line

! 1

average time spent waiting in line

1 average time in system including service

average number of customers in system

s

q

q q

q

PL

s

W L

W W

L W

r

r

Find Value for P0 from Chart Handout

Example: Multiple Server

• Computer Lab Help Desk• Now 45 students/hour need help.• 3 servers, each with service rate of 18

students/hour• Based on this, we know:

– µ = 18 students/hour– s = 3 servers– = 45 students/hour

Finding P0

r = 45/(3*18) =0.83

P0 ≈ 0.04

Probability of n Students in the System

Probability of Number in System

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.16000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number in System

Prob

abili

ty

Changing System Performance• 1. Customer Arrival Rates

– Ex:

• 2. Number and type of service facilities– Ex.

• 3. Change Number of Phases– Ex.

Changing System Performance• 4. Server efficiency

– Ex:

– Ex:

• 5. Change priority rules – Ex:

• 6. Change the number of lines– Ex:– Ex: