1. Queuing Systems
Chapter 17

2. Chapter Topics Elements of Waiting Line Analysis
Single-Server Waiting Line System
Undefined and Constant Service Times
Finite Queue Length
Finite Calling Problem
Multiple-Server Waiting Line
Addition Types of Queuing Systems

3. Overview
People and products spent significant amount of time in waiting lines
Providing quick service is an important aspect of customer service
Trade-off between the cost of improving service and the costs associated with making customers wait
A probabilistic form of analysis
Results are referred to as operating characteristics
Results are used to make decisions

4. Elements of Waiting Line Analysis
Waiting lines form because people arrive at a service faster than they can be served
Not arrive at a constant rate nor are they served in an equal amount of time
Have an average rate of customer arrivals and an average service time
Decisions are based on these averages for customer arrivals and service times
Formulas are used to compute operating characteristics

5. Single-Server Waiting Line System
Components include:
Arrivals (customers), servers, (cash register/operator), customers in line form a waiting line
Factors to consider:
queue discipline.
nature of the calling population
arrival rate
service rate.

6. Component Definitions
Queue Discipline: order in which customers are served (FIFO, LIFO, or randomly)
Calling Population: source of customers (infinite or finite)
Arrival Rate: Frequency at which customers arrive at a waiting line according to a probability distribution (Poisson distribution)
Service Rate: Average number of customers that can be served during a time period (negative exponential distribution)

7. Single-Server Waiting Line System Single-Server Model
Assumptions:
An infinite calling population
A first-come, first-served queue discipline
Poisson arrival rate
Exponential service times
Symbology:
 = the arrival rate (average number of arrivals/time period)
 = the service rate (average number served/time period)
Customers must be served faster than they arrive ( < )

8. Single-Server Queuing Formulas
Probability that no customers are in the waiting line:
Probability that n customers are in the waiting line:
Average number of customers in system: and waiting line:

9. SSingle-Server Queuing Formulas
Average time customer spends waiting and being served:
Average time customer spends waiting in the queue:
Probability that server is busy (utilization factor):
Probability that server is idle:

10. Characteristics for Fast Shop Market
 = 24 customers per hour arrive at checkout counter
 = 30 customers per hour can be checked out

11. Characteristics for Fast Shop Market

12. Steady-State Operating Characteristics
Utilization factor, U, must be less than one:
U < 1,or  /  < 1 and  < .
Ratio of the arrival rate to the service rate must be less than one or, the service rate must be greater than the arrival rate
Server must be able to serve customers faster than the arrival rate in the long run, or waiting line will grow to infinite size.

13. Single-Server Waiting Line System
Effect of Operating Characteristics (1 of 6)
Wish to test several alternatives for reducing customer waiting time:
Addition of another employee to pack up purchases
Addition of another checkout counter.
Alternative 1: Addition of an employee raises service rate from  = 30 to  = 40 customers
Cost $150 per week, avoids loss of $75 per week for each minute of reduced customer waiting time
System operating characteristics with new parameters:
Po = .40 probability of no customers in the system
L = 1.5 customers on the average in the queuing system

14. Effect of Operating Characteristics
System operating characteristics with new parameters (continued):
Lq = 0.90 customer on the average in the waiting line
W = hour average time in the system per customer
Wq = hour average time in the waiting line per customer
U = .60 probability that server is busy and customer must wait
I = .40 probability that server is available
Average customer waiting time reduced from 8 to 2.25 minutes worth $ =(8-2.25)($75) per week.

15. Effect of Operating Characteristics
Alternative 2: Addition of a new checkout counter ($6,000 plus $200 per week for additional cashier).
 = 24/2 = 12 customers per hour per checkout counter
 = 30 customers per hour at each counter
System operating characteristics with new parameters:
Po = .60 probability of no customers in the system
L = 0.67 customer in the queuing system
Lq = 0.27 customer in the waiting line
W = hour per customer in the system
Wq = hour per customer in the waiting line
U = .40 probability that a customer must wait
I = .60 probability that server is idle

16. Effect of Operating Characteristics
Savings from reduced waiting time worth $500= (8-1.33)($75) per week - $200 = $300 net savings per week.
After $6,000 recovered, alternative 2 would provide $ = $18.75 more savings per week.

17. Undefined and Constant Service Times
Constant occurs with machinery and automated equipment
Constant service times are a special case of the single-server model with undefined service times
Queuing formulas:

18. Undefined Service Times Example
Data: Arrival rate of 20 customers per hour (Poisson distributed); undefined service time with mean of 2 minutes, standard deviation of 4 minutes.
Operating characteristics:

19. Undefined Service Times Example (2 of 2)
Operating characteristics (continued):

20. Constant Service Times Formulas
No variability in service times;  = 0.
Substituting  = 0 into equations:
All remaining formulas are the same

21. Constant Service Times Example
Inspecting one car at a time; constant service time of 4.5 minutes; arrival rate of customers of 10 per hour (Poisson distributed).
Determine average length of waiting line and average waiting time
 = 10 cars per hour,  = 60/4.5 = 13.3 cars per hour

22. Finite Queue Length Length of the queue is limited.
Operating characteristics:
M is the maximum number in the system:

23. Finite Queue Length Example
Limited parking space (one vehicle in service and three waiting for service)
Mean time between arrivals of customers is 3 minutes and mean service time is 2 minutes (both inter-arrival times and service times are exponentially distributed)
Maximum number of vehicles in the system equals 4.
Operating characteristics for  = 20,  = 30, M = 4:

24. Example-Cont.
Average queue lengths and waiting times:

25. Finite Calling Population
There is a limited number of potential customers that can call on the system.
Operating characteristics (Poisson arrival and exponential service times):

26. Finite Calling Population Example
20 machines; each machine operates an average of 200 hours before breaking down; average time to repair is 3.6 hours; breakdown rate is Poisson distributed, service time is exponentially distributed.
Is repair staff sufficient?
 = 1/200 hour = .005 per hour
 = 1/3.6 hour = per hour
N = 20 machines

27. Finite Calling Population Example
System seems inadequate.

28. Multiple-Server Waiting Line
Two or more independent servers in parallel serve a single waiting line; first-come, first-served basis
Assumptions:
First-come first-served queue discipline
Poisson arrivals, exponential service times
Infinite calling population.
Parameter definitions:
 = arrival rate (average number of arrivals per time period)
 = the service rate (average number served per time period) per server (channel)
c = number of servers
c  = mean effective service rate for the system (must exceed arrival rate)

29. Multiple-Server Waiting Line: Formulas

30. Example
 = 10,  = 4, c = 3

31. Notation Used for Waiting Line Models
Kendal suggested A/B/k
A: Denotes the prob. Distribution for the arrival
B: Denotes the prob. Distribution for the service time
k: Denotes of channels
Letters commonly used
M: designates a Poisson prob. Dist. For the arrivals or an exp. prob. dist. for the service time
D: designates that arrivals or the service time is deterministic
G: designates that the arrivals or the service time has a general prob. Dist. With a known mean and variance
M/M/1: single-server with Poisson arrivals and exp. service time
M/M/2: two-server with Poisson arrivals and exp. service time
M/G/1: single-server with Poisson arrivals and arbitrary service time

32. Problem#1
A single server queuing system with an infinite calling population and a FIFO discipline has the following arrival and service rates:
l=16 customers/hour
m=24 customers/hour
Determine P0, P3, L, Lq, W, Wq, and U