1. Waiting Line Analysis for Service Improvement
Chapter 17
Waiting Line Analysis for Service Improvement
Operations Management - 5th Edition
Roberta Russell & Bernard W. Taylor, III

2. Lecture Outline What did we talk about last time?
Single Server Model
Operating Characteristics
Ways to measure the performance of a system
Variations on the Single Server Model
Multiple Server Model Analysis

3. Oh, the Joy of Lines Where do you encounter lines?
What makes a line form?
What is waiting line analysis?
What basic trade-off are we making?
Are lines avoidable?
Do we always want to avoid lines for
our service? Why or why not?

4. Operating Characteristics
NOTATION OPERATING CHARACTERISTIC
L Average number of customers in the system (waiting and being served)
Lq Average number of customers in the waiting line
W Average time a customer spends in the system (waiting and being served)
Wq Average time a customer spends waiting in line
P0 Probability of no (zero) customers in the system
Pn Probability of n customers in the system
ρ Utilization rate; the proportion of time the system is in use
Table 16.1

5. Basic Single-Server Model: Assumptions
Poisson arrival rate
Exponential service times
First-come, first-served queue discipline
Infinite queue length
Infinite calling population
 = mean arrival rate
 = mean service rate

6. Constant Service Times
Constant service times occur with machinery and automated equipment
Constant service times are a special case of the single-server model with undefined service times

7. Operating Characteristics for Constant Service Times 
P0 = 1 -
Probability that no customers
are in system
Average number of
customers in queue
Lq = 2
2( - )
Average number of
customers in system
L = Lq +  

8. Operating Characteristics for Constant Service Times (cont.)
Average time customer
spends in queue
Wq =
 Lq 
Average time customer
spends in the system
W = Wq + 1   
 =
Probability that the server is busy

9. Constant Service Times: Example
Automated car wash with service time = 4.5 min
Cars arrive at rate  = 10/hour (Poisson)
 = 60/4.5 = 13 1/3 per hour
(10)2
2(13.33)( )
Lq = = = cars waiting 2
2( - )
Wq = = 1.125/10 = hour or 6.75 min Lq 

10. Finite Queue Length A physical limit exists on length of queue
M = maximum number in queue
Service rate does not have to exceed arrival rate () to obtain steady-state conditions

11. Basic Multiple-server Model
Two or more independent servers serve a single waiting line
Poisson arrivals, exponential service, infinite calling population
s>
P0 = 1 s!   s s
s -  
n=s-1
n=0 n! n +
Computing P0 can be time-consuming.
Tables can used to find P0 for selected values of  and s.
Or use the OM Tools macro …

12. Basic Multiple-server Model (cont.)
Probability of exactly n customers in the system
Pn =
P0, for n > s 1
s! sn-s   n
P0, for n < s n!
Probability an arriving
customer must wait
Pw = P0 1 s! s
s -  s  
Average number of
customers in system
L = P0 +
(/)s
(s - 1)!(s - )2  

13. Basic Multiple-server Model (cont.)
W = L 
Average time customer
spends in system
Lq = L -  
Average number of
customers in queue
Average time customer
spends in queue
Wq = W = 1  Lq 
 = /s
Utilization factor

14. Multiple-Server System: Example
Student Health Service Waiting Room
 = 10 students per hour
 = 4 students per hour per service representative
s = 3 nurses
s = (3)(4) = 12
P0 =
Probability no students are in the system
Number of students in the service area
L =

15. Multiple-Server System: Example (cont.)
Waiting time in the service area
W = L / l = min
Lq = L - l/m =
Number of students waiting to be served
Average time students will wait in line
Wq = Lq/l = min
Probability that a student must wait
Pw =

16. Add Another Nurse? Students are frustrated with 21 minute wait
It would cost $25,000 a year to add another nurse
Each minute reduced for the students’ waiting time in line saves the health center an estimated $1500

17. Multiple-Server System: Example (cont.)
Add a 4th server to improve service
Recompute operating characteristics
P0 = prob of no students
L = students
W = min in service
Lq = students waiting
Wq = min waiting (versus 21 earlier)
Pw = prob that a student must wait