Chapter 16

Capacity Planning and

Queuing Models

Terminology Capacity

is the ability to deliver service over a particular time

period.

is determined by the resources available to the

organization in the form of facilities, equipment and labor.

Capacity Planning

is the process of determining the types and amounts of

resources that are required to implement an

organization’s strategic business plan.

is a challenge for service firms because of the open

system nature of service operations.

Keynotes

The capacity planning decision involves a trade-off

between the cost of providing a service and the cost of

inconvenience of customer waiting.

The cost of service capacity is determined by the

number of servers on duty, whereas customer

inconvenience is measured by waiting time.

The lack of control over customer demands for service

and the presence of the customer in the process

complicate capacity planning.

For services, it is necessary to predict the degree of

customer waiting associated with different levels of

capacity.

14-4

Capacity Planning Challenges

Inability to create a steady flow of demand to fully utilize

capacity

Idle capacity always a reality for services.

Customer arrivals fluctuate and service demands also

vary.

Customers are participants in the service and the level of

congestion impacts on perceived quality.

Inability to control demand results in capacity measured in

terms of inputs (e.g. number of hotel rooms rather than

guest nights).

14-5

Strategic Role of Capacity Decisions

Capacity decision in services have strategic importance

based on the time horizon.

Lack of short-term capacity planning can generate

customers for competition (e.g. restaurant staffing)

Capacity decisions that must be balanced against the

costs of lost sales if capacity is inadequate or against

operating losses if demand does not reach expectations.

Strategy of building ahead of demand is often taken to

avoid losing customers.

Queuing System Cost Tradeoff

Let: Cw = Cost of one customer waiting in queue for an hour

Cs = Hourly cost per server

C = Number of servers

Total Cost/hour = Hourly Service Cost + Hourly Customer Waiting Cost

Total Cost/hour = Cs C + Cw Lq

Note: Only consider systems where

C

16-6

Analytical Queuing Models

A popular system classifies parallel-server queuing models

using A / B / C.

A represents the distribution of time between arrivals.

B represents the distribution of service times

C represents the number of parallel servers.

The descriptive symbols used for the arrival and service

distributions include

M= exponential interarrival or service time distribution (or

Poisson distribution of arrival or service rate)

D= deterministic or constant interarrival or service time

Ek = Earlang distribution with shape parameter k (if k=1 then

exponential; if k= ∞ then deterministic)

G= general distribution with mean and variance (normal,

uniform or any empirical distribution)

M / M / 1 = a single server queuing model with Poisson

arrival rate and exponential service time distribution.

Classification of Queuing Models

Standard

(Infinite Queue)

Queuing Models

Poisson Arrivals

Finite Queue

Exponential

Service Times

General

Service Times

Exponential

Service Times

VI

Multiple

Servers

M / M /c

V

Single

Server

M / M /1

IV

Self

Service

M / G /∞

III

Single

Server

M / G /1

II

Multiple

Servers

M / M /c

I

Single

Server

M / M /1

Notation in Equations n= number of customers in the systems

λ = mean arrival rate

μ = mean service rate per busy server

ρ = (λ / μ) mean number of customers in service

N = max number of customers allowed in the system

c = number of servers

Pn = probability of exactly n customers in the system

Ls = mean number of customers in the system

Lq = mean number of customers in queue

Lb = mean number of customers in queue for a busy system

Ws = mean number of customers in the system

Wq = mean number of customers in queue

Wb = mean number of customers in queue for a busy system

Standard M / M / 1 Model Assumptions

Calling Population: An infinite or very large population of

callers arriving. The callers are independent of each other

and not influenced by the queuing system.

Arrival process: Negative exponential distribution of

interarrival times or Poisson distribution of arrival rate.

Queue configuration: Single waiting line with no restrictions

on length and no balking or reneging.

Queue discipline: First come first serve (FCFS)

Service Process: One server with negative exponential

distribution of service times.

14-12

Queuing Formulas

Single Server Model with Poisson Arrival and Service Rates: M/M/1

1. Mean arrival rate:

2. Mean service rate:

3. Mean number in service:

4. Probability of exactly “n” customers in the system:

5. Probability of “k” or more customers in the system:

6. Mean number of customers in the system:

7. Mean number of customers in queue:

8. Mean time in system:

9. Mean time in queue:

Pn

n ( )1

P n k k( )

Ls

Lq

Ws

1

Wq

14-13

Queuing Formulas (cont.)

Single Server General Service Distribution Model: M/G/1

Mean number of customers in queue for two servers: M/M/2

Relationships among system characteristics:

Lq

2 2 2

2 1( )

Lq

3

24

Ls Lq

Ws Wq

Ws Ls

Wq Lq

1

1

1

Congestion as 10.

0 1.0

100

10

8

6

4

2

0

With:

Ls1

Then:

Ls

0 0

0.2 0.25

0.5 1

0.8 4

0.9 9

0.99 99

16-14

General Queuing Observations

1. Variability in arrivals and service times contribute equally to

congestion as measured by Lq.

2. Service capacity must exceed demand.

3. Servers must be idle some of the time.

4. Single queue preferred to multiple queue unless jockeying

is permitted.

5. Large single server (team) preferred to multiple-servers if

minimizing mean time in system, WS.

6. Multiple-servers preferred to single large server (team) if

minimizing mean time in queue, WQ.

16-15

Appendix D

Equations for selected queuing models

1. Standard M/M/1 Model (0<ρ<1.0)

2. Standard M/M/c Model (0<ρ<c)

3. Standard M/G/1 Model (V(t)=service time

variance)

4. Self-service M/G/∞ Model

5. Finite-Queue M/M/1 Model

6. Finite-Queue M/M/c Model

1. Standard M/M/1 Model (0<ρ<1.0)

1. Calling population. An infinite or very large population of callers

arriving. The callers are independent of each other and not

influenced by the queuing system.

2. Arrival process. Poisson distribution of arrival rate.

3. Queue configuration. Single waiting line with no restrictions on

length and no balking or reneging.

4. Queue discipline. FIFO

5. Service process. One server with exponential distribution of

service times.

Example 16.2.

λ=6 boats per hour (poisson)

μ=6 minutes per boat (10 boat per hour) (exponential)

M/M/1 model (infinite population, no queue length restrictions, no balking or reneging, and FCFS queue discipline)

λ Single server

Ls

Lq

μ

ρ=λ/μ=6/10= 0.60

Probability that the system is busy and an arriving customer waits: P(n≥1)= ρ1=0.601=0.60

Probability of finding the ramp idle : P0=1- ρ=1-0.60=0.40

Mean number of boats in the system:

Ls

k)kn(P

boats 5.1610

6Ls

Mean number of boats in queue:

Lq

boat 90.0610

)6)(60.0(Lq

Mean time in the system:

1

Ws

min.) (15 hour 25.0610

11Ws

Mean time in queue:

min.) (9 hour 15.0610

60.0Wq

Wq

The boat ramp is busy 60 % of the time. ◦ Thus, arrivals can expect immediate access to the ramp without delay

40% of the time.

The mean time in the system of 15 minutes is the sum of the mean time in queue of 9 minutes and the mean service time of 6 minutes.

Arrivals an expect to find the number in the system to be 1.5 boats and the expected number in queue to be 0.9 boat.

The number of customers in the system can be used to identify system states. ◦ For example, when n=0, the system is idle.

◦ When n=1, the server is busy but no queue exists.

◦ When n=2, the server is busy and a queue of 1 has formed.

◦ The probability distribution for n can be very uaeful in determining the proper size of a waiting room (i.e., the number of chairs) to accommodate arriving customers with a certain probability of assurance that each will find a vacant chair.

For the boot ramp example, determine the number of parking spaces needed to ensure that 90 % of the time, a person arriving at the boot ramp will find a space to park while waiting to launch.

n Pn P (number of customers ≤ n)

0 (0.6)0(0.4)=0.40 0.40

1 (0.6)1(0.4)=0.24 0.64

2 (0.6)2(0.4)=0.144 0.784

3 (0.6)3(0.4)=0.0864 0.8704

4 (0.6)4(0.4)=0.05184 0.92224

)1(P nn

Repeatedly using the probability distribution for system states for increasing values of n, we accumulate the system state probabilities until 90 % assurance is exceeded.

A system state of n=4 or less will occur 92% of the time.

This suggests that room for 4 boat trailers should be provided because 92% of the time arrivals will find 3 or fewer people waiting in queue to launch.

EXAMPLE

A Social Security Administration branch is considering the following two

options for processing applications for social security cards:

◦ Option 1: Three clerks process applications in parallel from a single

queue. Each clerk fills out the form for the application in the presence of

the applicant. Processing time is exponential with a mean of 15 minutes.

Interarrival times are exponential.

◦ Option 2: Each applicant first fills out an application without the clerk’s

help. The time to accomplish this is exponentially distributed, with a

mean of 65 minutes. When the applicant has filled out the form, he or

she joins a single line to wait for one of the three clerks to check the

form. It takes a clerk an average of 4 minutes (exponentially distributed)

to review an application.

The interarrival time of applicants is exponential, and an average of 4.8

applicants arrive each hour.

Which option will get applicants out of the more quickly?

For Option 1

Option 1 is an M/M/c system with λ = 4.8 applicants/hr. and µ = 4 applicants/hour.

c=3 and ρ = 4.8/4 = 1.2 , from table Lq = 0.094 applicants

Ls=Lq + ρ = 0.094 + 1.2 = 1.294 applicants

Ws = Ls/λ = 1.294/4.8 = 0 .27 hours

EXAMPLE

Last National Bank is concerned about the level of service at its single drive-in window. A study of customer arrivals during the window’s busy period revealed that, on average, 20 customers per hour arrive, with a Poisson distribution, and they are given FCFS service, requiring an average of 2 minutes, with service times having an exponential distribution.