1. LESSONs NINE and TEN
QUEUING
MODELS

2. Queuing Models Queuing is the study of waiting lines, or queues.
The objective of queuing analysis is to design systems that enable organizations to perform optimally according to some criterion.
Possible Criteria
Maximum Profits.
Desired Service Level.

3. Elements of the queuing process
A queuing system consists of five basic components:
Calling population
Arrival process
Queue configuration
Queue discipline
Service process

4. CALLING POPULATION

5. CALLING POPULATION
Homogeneous or heterogeneous
Can be finite by limiting calls or infinite if population is big

6. ARRIVAL PROCESS

7. ARRIVAL PROCESS
Static no control
Dynamic control by machines or man
Balking not joining the queuing
Reneging leave line before service

8. QUEUE CONFIGURATION

9. QUEUE CONFIGURATION
Refers to number of queues, locations, spatial requirement and effect on customer behavior
Jockeying – line switching activity

10. QUEUE DISCIPLINE

11. SERVICE PROCESS

12. Possible Service Measurements
Average time a customer spends in line.
Average length of the waiting line.
The probability that an arriving customer must wait for service.

13. The Arrival Process The random process is more common in businesses.
Under three conditions a Poisson Distribution can describe the random arrival process.

14. The three conditions required for the existence of the Poisson arrival process:
Orderliness : one customer, at most, will arrive during any time interval.
Stationarity : for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length.
Independence : the arrival of one customer has no influence on the arrival of another.
These conditions are unrestrictive and are approximately satisfied in many situations.

15. The Poisson Arrival DistributionX k e ! = l t) t ( ) -
Where
l = mean arrival rate per time unit.
t = the length of the interval.
e = (the base of the natural logarithm).
k! = k (k -1) (k -2) (k -3) … (3) (2) (1).

16. HANK’s HARDWARE (An illustration of the Poisson distribution)
Customers arrive at Hank’s Hardware according to a Poisson distribution.
Between 8:00 a.m. and 9:00 a.m., an average of 6 customers arrive at the store.
What is the probability that k = 0, 1, 2, … customers will arrive between 8:00 and 8:30 in the morning.

17. SOLUTION 1 2 3 Input to the Poisson distribution
SOLUTION
Input to the Poisson distribution
l= 6 customers per hour.
t = 0.5 hour.
l t = (6)(0.5) = 3. 1 2 3 4 5 6 7 8 3 2 1 1 = 2 3 0! 3! 2! 1!

18. Measures of Queuing System Performance
P0 = Probability that there are no customers in the
system.
Pn = Probability that there are “n” customers in the
L = Average number of customers in the system.
Lq = Average number of customers in the queue.
W = Average time a customer spends in the system.
Wq = Average time a customer spends in the queue.
Pw = Probability that an arriving customer must wait for service.
r = Utilization rate for each server (the percentage of time that each server is busy).

19. Classification of Queues Queuing system can be classified by:
Arrival process.
Service process.
Number of servers.
System size (infinite/finite waiting line).
Population size.
Notation
M (Markovian) = Poisson arrivals or exponential service time.
D (Deterministic) = Constant arrival rate or service time.
G (General) = General probability for arrivals or service time.
Example:
M / M / 6 / 10 / 20

20. M / M / 1 Queuing System Characteristics Poisson arrival process.
Exponential service time distribution.
A single server.
Potentially infinite queue.
An infinite population.

21. Performance Measures for the M / M /1 Queue
P0 = 1- (l / m)
Pn = [1 - (l / m)] (l/ m)n
L = l / (m - l)
Lq = l 2 / [m(m - l)]
W = 1 / (m - l)
Wq = l / [m(m - l)]
Pw = l / m
r = l / m

22. MARY’s SHOES
Customers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process.
Service time is exponentially distributed with an average of 8 minutes per customer.
Management is interested in determining the performance measures for this service system.

23. SOLUTION P0 = 1- (l / m) = 1 - (5 / 7.5) = 0.3333
Input
l = 1/ 12 customers per minute
= 60/ 12 = 5 per hour.
m = 1/ 8 customers per minute
= 60/ 8 = 7.5 per hour.
Performance Calculations
P0 = 1- (l / m) = 1 - (5 / 7.5) =
Pn = [1 - (l / m)] (l/ m) = (0.3333)(0.6667)n
L = l / (m - l) = 2
Lq = l2/ [m(m - l)] =
W = 1 / (m - l) = 0.4 hours = 24 minutes
Wq = l / [m(m - l)] = hours = 16 minutes
Pw = l / m =
r = l / m =

24. m l
WINQSB Input Screen

25. Performance Measurements

26. M / M / k Queuing Systems Characteristics
Customers arrive according to a Poisson process at a mean rate l.
Service time follow an exponential distribution.
There are k servers, each of which works at a rate of m customers.
Infinite population, and possibly infinite line.

27. Performance Measure

28. The performance measurements L, Lq, Wq,, can be obtained
from Little’s formulas.

29. Revision Question 
At a Food Lion Provision Shop, customers spend an average of 25 mins selecting their groceries and checking out by entering a single line queue served by two cashiers. The service times required for the cashiers to check out customers follow an exponential distribution and average four minutes. Customers arrive at the cashier counter according to a Poisson distribution at the average rate of eight customers per hour The table below shows part of the computer calculation:
Determine the followings :
average time a customer spends in the store (in minutes).
average number of customers waiting in line prior to being checked out.
proportion of customers who will have to wait in line.  
What assumption(s) did you make in part (a)? Comment whether the assumption(s) is/are realistic.

30. Queuing Process Multiple queue advantages :
Service can be differentiated
Division of labour possible
Selection option for customer
Deterred balking

31. Types of services clearly stated
Queuing Process
Multiple queue disadvantages :
Anxiety
Lack of fairness
Lack of privacy
Types of services clearly stated

32. Queuing Process Single queue advantages :
First come first serve (fairness)
No anxiety to select fastest line
Reneging difficult
Queue cutting resolved
Privacy enhanced
Reducing average waiting time

33. Unnecessary held up in waiting line
Queuing Process
Single queue disadvantages :
No specialisation
Unnecessary held up in waiting line
Possible balking

34. Free to wander about and browse items
Queuing Process
Take a number advantages :
No need for formal line
Free to wander about and browse items
More relaxing

35. Require large waiting area
Queuing Process
Take a number disadvantages :
Reneging
Require large waiting area

36. Queuing Process QUEUE DISCIPLINE
Policy of selecting next customer from the queue for service
First come first serve most popular static method (selection depends on position in queue only)

37. Queuing Process
Dynamic disciplines selection involve some attribute or status in selection
Shortest processing time minimise average time of customers

38. Queuing Process
Preemptive priority interrupt current person service for newly arrived customer with higher priority e.g. fire or ambulance services
Round robin service give customer partial service and then movers to next waiting customer – alternating between waiting and being served

39. Queuing Process
Service person begins to take orders while customers are still waiting in line is a direct approach to avoid reneging

40. Queuing Process SERVICE PROCESS
Distribution of service times, arrangement of servers, management policies and server behaviour contribute to performance

41. WAITING PERCEPTION Unoccupied time feels longer than occupied time
Preprocess waits feel longer than in process waits
Anxiety makes waits seem longer

42. WAITING PERCEPTION Uncertain waits are longer than known finite waits
Unexplained waits are longer than explained wait
Unfair waits are longer than equitable waits

43. WAITING PERCEPTION People more willing to wait for valuable service
Solo waiting feels longer than group waiting
Customer attitudes

44. WAITING PERCEPTION Environment
Unused facilities and idle staff increase annoyance
Unfamiliar music makes perceived time seem longer than familiar background music

45. REVIEW QUESTIONS 1. What does M/D/1 refers to?
2. What are the characteristics of
M / G / 1?
3. How can you measure the performance of a queue?

46. REVIEW QUESTIONS
4. List and discuss the factors affecting queue perception.
5. What are the five features of a queue?
6. How can you configure queue?