1. Operations Management
Module D – Waiting-Line Models
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 6e
Operations Management, 8e
© 2006 Prentice Hall, Inc.

2. Common Queuing Situations
Arrivals in Queue
Service Process
Supermarket
Grocery shoppers
Checkout clerks at cash register
Highway toll booth
Automobiles
Collection of tolls at booth
Doctor’s office
Patients
Treatment by doctors and nurses
Computer system
Programs to be run
Computer processes jobs
Telephone company
Callers
Switching equipment to forward calls
Bank
Customer
Transactions handled by teller
Machine maintenance
Broken machines
Repair people fix machines
Harbor
Ships and barges
Dock workers load and unload
Table D.1

3. Characteristics of Waiting-Line Systems
Arrivals or inputs to the system
Population size, behavior, statistical distribution
Queue discipline, or the waiting line itself
Limited or unlimited in length, discipline of people or items in it
The service facility
Design, statistical distribution of service times
There are 3 parts of a waiting line, or queuing, system. See the next slide for a nice picture.

4. Parts of a Waiting Line Arrival Characteristics Size of the population
Population of
dirty cars
Arrivals
from the
general
population …
Queue
(waiting line)
Service
facility
Exit the system
Dave’s
Car Wash
enter
exit
Arrivals to the system
In the system
Picture of the three parts of the queue that we are interested in to help characterize it.
Exit the system
Arrival Characteristics
Size of the population
Behavior of arrivals
Statistical distribution of arrivals
Waiting Line Characteristics
Limited vs. unlimited
Queue discipline
Service Characteristics
Service design
Statistical distribution of service
Figure D.1

5. Arrival Characteristics
Size of the population
Unlimited (infinite) or limited (finite)
Pattern of arrivals (statistical distribution)
Scheduled or random, often a Poisson distribution
Behavior of arrivals
Wait in the queue and do not switch lines
Balking or reneging
First part is: Arrival Characteristics.
Difference between balking and reneging:
Balking – refusing to join the line (queue) because it is too long.
Reneging – enter the queue, but become inpatient and leave without being served.

6. Poisson Distribution e-x P(x) = for x = 0, 1, 2, 3, 4, … x!
where P(x) = probability of x arrivals
x = number of arrivals per unit of time
 = average arrival rate
e = (which is the base of the natural logarithms)
This distribution is assumed and used in much of queuing theory. Why do you think this is the case?

7. Poisson Distribution e-x Probability = P(x) = x! 0.25 – 0.25 –
0.25 –
0.20 –
0.15 –
0.10 –
0.05 – –
Probability 1 2 3 4 5 6 7 8 9
Distribution for  = 2 x
0.25 –
0.20 –
0.15 –
0.10 –
0.05 – –
Probability 1 2 3 4 5 6 7 8 9
Distribution for  = 4 x 10 11
Arrivals are not always Poisson distributed. Therefore, historical data should be examined carefully before applying any of the models that assume an underlying Poisson arrival process.
Figure D.2

8. Waiting-Line Characteristics
Limited or unlimited queue length
Queue discipline: first-in, first-out is most common
Other priority rules may be used in special circumstances
Second part is: Queue Characteristics
For all of the models we study in this Module, we will assume unlimited queue length.
First-in, first-out (abbreviated as FIFO) is very similar to the FCFS rule we studied in Chapter 15 for sequencing.

9. Service Characteristics
Queuing system designs
Single-channel system, multiple-channel system
Single-phase system, multiphase system
Service time distribution
Constant service time
Random service times, usually a negative exponential distribution
Third part: Service Characteristics
Two basic properties are important: (1) design of the service system and (2) the distribution of the service times.
Service systems are often classified in terms of their number of channels and the number of phases.

10. Queuing System Designs
Your family dentist’s office
Queue
Service facility
Departures
after service
Arrivals
Single-channel, single-phase system
McDonald’s dual window drive-through
We will look at four possible channel and phase configurations.
Queue
Phase 1 service facility
Phase 2 service facility
Departures
after service
Arrivals
Single-channel, multiphase system
Figure D.3

11. Queuing System Designs
Most bank and post office service windows
Service facility
Channel 1
Channel 2
Channel 3
Departures
after service
Queue
Arrivals
Multi-channel, single-phase system
Figure D.3

12. Queuing System Designs
Some college registrations
Phase 1 service facility
Channel 1
Channel 2
Phase 2 service facility
Channel 1
Channel 2
Queue
Departures
after service
Arrivals
Any other good examples?
Multi-channel, multiphase system
Figure D.3

13. Negative Exponential Distribution
Probability that service time is greater than t = e-µt for t ≥ 1
µ = Average service rate
e =
1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 –
Probability that service time ≥ 1
| | | | | | | | | | | | |
Time t in hours
Average service rate (µ) = 3 customers per hour
 Average service time = 20 minutes per customer
Service time distributions are similar to arrival time distributions: the can be constant (simplest case) or random.
If we assume a negative exponential distribution with a service rate of 3 customers/hour, then it will be very, very rare that any one person takes longer than 1.5 hours to serve.
Average service rate (µ) =
1 customer per hour
Figure D.4

14. Measuring Queue Performance
Average time that each customer or object spends in the queue
Average queue length
Average time in the system
Average number of customers in the system
Probability the service facility will be idle
Utilization factor for the system
Probability of a specified number of customers in the system
Metrics, lots and lots of metrics!

15. Cost of providing service
Queuing Costs
Cost
Low level
of service
High level
Cost of waiting time
Total expected cost
Minimum
Total
cost
Cost of providing service
When operating any service where a queue may exist, you have to balance the benefits between two basic trade-offs: (1) the cost of providing good service and (2) the cost of the customer (or machine) waiting time. One way to look at the problem is by looking at the total expected cost.
Optimal
service level
Figure D.5

16. Queuing Models Model Name Example A Single channel Information counter
system at department store
(M/M/1)
Number Number Arrival Service
of of Rate Time Population Queue
Channels Phases Pattern Pattern Size Discipline
Single Single Poisson Exponential Unlimited FIFO
The four models presented in Module B have three characteristics in common: (1) Poisson arrivals, (2) FIFO discipline, and (3) Single-service phase.
To describe this queuing model, the notation M/M/1 is used. The first letter refers to the arrival process (where M stands for Poisson). The second letter refers to the service (where M is again a Poisson distribution, which is the same as an exponential rate for service). The third symbol refers to the number of servers.
Table D.2

17. Queuing Models Model Name Example B Multichannel Airline ticket
(M/M/S) counter
Number Number Arrival Service
of of Rate Time Population Queue
Channels Phases Pattern Pattern Size Discipline
Multi- Single Poisson Exponential Unlimited FIFO
channel
Table D.2

18. Queuing Models Model Name Example C Constant Automated car
service wash
(M/D/1)
Number Number Arrival Service
of of Rate Time Population Queue
Channels Phases Pattern Pattern Size Discipline
Single Single Poisson Constant Unlimited FIFO
Table D.2

19. Queuing Models Model Name Example D Limited Shop with only a
population dozen machines
(finite) that might break
Number Number Arrival Service
of of Rate Time Population Queue
Channels Phases Pattern Pattern Size Discipline
Single Single Poisson Exponential Limited FIFO
Table D.2

20. Model A - Single Channel
Arrivals are FIFO and every arrival waits to be served regardless of the length of the queue
Arrivals are independent of preceding arrivals but the average number of arrivals does not change over time
Arrivals are described by a Poisson probability distribution and come from an infinite population
6 major assumptions with this model – here are the first three.

21. Model A - Single Channel
Service times vary from one customer to the next and are independent of one another, but their average rate is known
Service times occur according to the negative exponential distribution
The service rate is faster than the arrival rate
6 major assumptions with this model – here are the final three.
Why is assumption 6 important?

22. Model A - Single Channel
 = Mean number of arrivals per time period
µ = Mean number of units served per time period
Ls = Average number of units (customers) in the system (waiting and being served) =
Ws = Average time a unit spends in the system (waiting time plus service time) 
µ –  1
Of course, you have to put everything into the same time period to ensure the model makes sense.
Table D.3

23. Model A - Single Channel
Lq = Average number of units waiting in the queue =
Wq = Average time a unit spends waiting in the queue
ρ = Utilization factor for the system 2
µ(µ – ) 
Table D.3

24. Model A - Single Channel
P0 = Probability of 0 units in the system (that is, the service unit is idle)
= 1 –
Pn > k = Probability of more than k units in the system, where n is the number of units in the system = 
k + 1
Table D.3

25. Single Channel Example
 = 2 cars arriving/hour µ = 3 cars serviced/hour
Ls = = = 2 cars in the system on average
Ws = = = 1 hour average waiting time in the system
Lq = = = cars waiting in line 2
µ(µ – ) 
µ –  1 2
3 - 2 22
3(3 - 2)
Example from page 751 – Golden Muffler Shop.

26. Single Channel Example
 = 2 cars arriving/hour µ = 3 cars serviced/hour
Wq = = = 40 minute average waiting time
ρ = /µ = 2/3 = 66.6% of time mechanic is busy 
µ(µ – ) 2
3(3 - 2)
P0 = = .33 probability there are 0 cars in the system

27. Single Channel Example
Probability of More Than k Cars in the System
k Pn > k = (2/3)k + 1
 Note that this is equal to 1 - P0 =
1 .444
2 .296
 Implies that there is a 19.8% chance that more than 3 cars are in the system
4 .132
5 .088
6 .058
7 .039
NOTE: The OM macro does NOT perform these calculations for you.

28. Single Channel Economics
Customer dissatisfaction and lost goodwill = $10 per hour
Wq = 2/3 hour
Total arrivals = 16 per day
Mechanic’s salary = $56 per day
Total hours customers spend waiting per day
= (16) = hours 2 3
While having the characteristics of the queue is important, we may also want to talk in dollars and cents.
Customer waiting-time cost = $ = $106.67 2 3
Total expected costs = $ $56 = $162.67