1. Chapter 18 Management of Waiting Lines
Cheng Li, Ph.D.
Management Department
California State University, Los Angeles

2. Waiting Lines Examples of waiting lines
Car wash: three stages, multiple channels (servers) for vacuuming and drying, single channel for washing
A network printer: single stage, single channel
A maintenance worker: single channel, finite number of customers
Raw materials waiting to be processed

3. Elements of Queuing System: single phase, single channel
Arrivals
Service
Waiting
line
Exit
Processing
order
System

4. Elements of Queuing System: multiple phases, multiple channels, single line

5. Waiting Lines Queuing theory: mathematical analysis of waiting lines
Decision: capacity level that optimally balances the tradeoff between:
Customer waiting costs, and
Service capacity costs

6. Costs of Waiting Lines
Waiting lines: delay in work process (“inventory” holding cost)
Loss of business: customers refusing to wait
Loss of goodwill, decrease in customer satisfaction
Impact on other operations in the same process: disruptions caused by congestion at the bottleneck
Cost of waiting space

7. Cost Tradeoff Cost Service capacity Total cost Customer waiting cost= +
Total cost
Lowest TC
Cost of service capacity
Cost of customers
waiting
Service capacity

8. Waiting Lines
Assumption: no change in process design, but future improvement may benefit from the results of analysis
Roles of Queuing Analysis:
Estimate system performance for given capacity decisions
Provide input for future process improvement

9. System Characteristics
Customer population:
Infinite source: unlimited number of potential arrivals in a given period of time
Finite source: limited number of potential arrivals in a given period of time
Number of phases: distinguished by:
a designated server for each phase
a potential waiting line between consecutive phases

10. System Characteristics
Number of channels (servers): max. number of customers being served at the same time
Number of lines (multiple channels)
Single line vs. Multiple lines
Advantages of single lines
Less waiting
Higher utilization of capacity
Customer perception
Question: Why do some businesses still use multiple lines?

11. System Characteristics
Arrival patterns
Arrival rate: number of arrivals per period
probabilistic: Poisson distribution
deterministic: predetermined (appointment system)
Inter-arrival time: time between two consecutive arrivals, inverse of arrival rate
Service patterns: probabilistic vs. deterministic
Service Rate: inverse of service time
Service Time:
Probabilistic: exponential distribution
Deterministic: constant

12. Poisson & Exponential Distributions
Poisson: e.g. service rate
Exponential: e.g. service time
Poisson: mean = 4 (e.g. 4 customers per hour)
Exponential: lamda = 1 (parameter value)
Service time for the exponential distribution is inverse of service rate values. For instance: 2 customers per hour => 0.5 hours of service time per customer; 4 hours of service time per customer => 0.25 customers per hour.

13. System Characteristics
Queue discipline: priority rules used to determine the order of customers
FCFS: First Come, First Server
EDD: Earliest Due Date
SPT: Shortest Processing Time
Appointment system

14. System Performance: Basic Measures
Capacity measures
Capacity utilization
Average number of customers waiting
Average number of customers in system (waiting + being served)
Customer measures
Average time customers wait
Average service time
Average time in system

15. Performance Measurement for Queuing Models: Infinite Source
Single channel, exponential service time
Single channel, constant service time
Multiple channel, exponential service time
Other assumptions:
Single phase
Poisson arrival distribution (= exponential distribution for inter-arrival time)
FCFS

16. Infinite Source: General Formulas
System Utilization
Idle Rate
Avg. # of customers in line
Avg. # of customers in system
Avg. waiting time
Avg. service time
Avg. time in system

17. Infinite Source: single channel
Avg. # of customers in line
Prob.{0 customer in system}
Prob.{n customers in system}
Prob.{customers in system<n}

18. Infinite Source: constant service time
Avg. # of customers in line
Eliminate variability in service time
Reduce waiting line/time by half

19. System Performance: Other Measures
Question: Is “average” a good measure?
HALF of the total is worse than average (for a symmetric distribution)
Does not tell about the worst cases
Does not tell about spread of distribution
SD=2: 12.1%
SD=1: 5.4%

20. Infinite Source: single channel
Probability regarding # of customers in system:

21. Infinite Source: single channel
Probability regarding # of customers in system:

22. System Performance: Application of Measures
Probability that an arrival will have to wait (1-P0)
% of callers who are put on hold
Probability of more than a given number of customers in line
% of callers who have to wait for more than a given amount of time
Probability that number of customers waiting will exceed a given number
Chance of exceeding the capacity limit of the waiting area

23. System Performance Implied cost Implied constraints
Capacity costs: e.g. #servers x wage
Customer related costs: e.g. costs associated with a given level of waiting
Space: unlimited space assumed, but: “Will space limit be exceeded?”
Implied constraints
Business process: how service is divided into phases, tasks at each phase, etc.

24. Utilization & Waiting:20 60

25. Queuing Models Single channel, exponential service time
Single channel, constant service time
Multiple channel, exponential service time
Multiple priority service

26. Priority Model Arrivals Service Waiting line Exit Processing order
System 1 2 3
Arrivals are assigned
a priority as they arrive

27. Finite-Source Formulas
Table 19-6
Average number being served
Service factor
Average number waiting
Average waiting time
Average number running
Number in population

28. Finite-Source Queuing
Not waiting or
being served
Waiting
Being
served J L H U W T