1. Waiting Lines Students should be able to:
Categorize a waiting line problem as to customer population, waiting line arrangement, and service facility arrangement.
Describe the operating characteristics of waiting line systems and the important decisions.
Apply the single-server and multiple-server models.
Use Little’s Law to estimate the number of customers in the service system.

2. Waiting Line Models Customer population Service system Waiting line
Served customers
Waiting line
Priority rule
Service facilities
Figure C.1

3. Waiting Line Arrangements
Service facilities
(a) Single line
Waiting Line Arrangements
Service facilities
(b) Multiple lines
Figure C.2

4. Operating Characteristics
Line length
Number of customers in system
Waiting time in line
Total time in system
Service facility utilization

5. Decision Areas Arrival rates Number of service facilities
Number of phases
Number of servers per facility
Server efficiency
Priority rule
Service facility arrangement

6. Waiting Line Assumptions
Customer population : Infinite and patient
Customer arrivals : Poisson distribution
Mean arrival rate = l
3. Service times : Exponential distribution
Mean service rate = m
Priority rule : First-come-first-served (FCFS)
Waiting line : Unlimited length

7. Service Facility Arrangements
Single channel, Single phase

8. Single-Server Model Single-Channel, Single-Phase System
Arrival rate = 30/hour Service rate = 35/hour l m 30 35
Utilization =  = = = 0.857, or 85.7%
m - l 30
35 – 30 l
Average number in system = L = = = 6 customers
Average number in line = Lq =  L = 0.857(6) = 5.14 customers 1
m - l 1
35 – 30
Average time in system = W = = =
0.20 hour
Average time in line = Wq =  W = 0.857(0.20) = 0.17 hour
Example C.3

9. Little’s Law

10. Service Facility Arrangements
Multiple channel, Single phase

11. Multiple-Server Model Multiple-Channel, Single-Phase System
Distribution Center
s = 4 unloading bays
λ = 3 trucks arrive per hour
μ = 1 truck unloaded per hour per bay
Suppose that it was observed that the average number of trucks waiting to be unloaded is trucks.
hours waiting
hours being unloaded and waiting
trucks in the system
utilization

12. Designing a Distribution Center
Proposed design: 4 unloading bays
Employees required: 2 per bay -- $30 per hour each
Truck arrivals: 3 per hour
Unloading rate: 1 truck per hour per team
Idle truck cost: $100 per hour
Number Number of Employee Number of Truck
of Bays Employees Cost/Hr Trucks in System Cost/Hr Total
$ $ $693 $ $ $
5 Bays are the least expensive to operate.

13. Choosing a Service Facility Arrangement
Suppose you have two servers, each can service 5 customers per hour.
In Design 1, which is two single-line, single-server facilities, each server has her own line and each receives and average of 4 customers per hour.
In Design 2, which is a single-line, multiple-server facility, there are two servers and an average demand of 8 customers per hour.
How do the designs affect customer service?

14. How Does Facility Arrangement Affect Customer Service?
Design Design 2
Utilization
Number in System
Number in Line
Wait Time in System
Wait Time in Line

15. NYPD Patrol System Data The average service time is 30 minutes.
Precinct statistics
Precinct Average Calls Total Felonies Size
per Hour per Year Sq. Miles
________ ___________ ___________ _______
A ,
B ,
If you had a total of 9 cars to allocate, how would you do it?

16. Precinct Waiting Statistics
Precinct A
# Cars
Utilization 4 5 6 7 8
.373
.080
.023
.007
.002
2.390
.513
.145
. 043
. 012
.800
.640
.533
.457
.400
Precinct B
# Cars
Utilization 2 3 4 5 6