The document discusses queuing theory and waiting lines. Queuing theory uses a mathematical approach to analyze waiting lines. The goal is to minimize the sum of customer waiting costs and service capacity costs. Key factors of queuing systems include arrival and service patterns, population source, number of servers, and queue discipline. Performance is measured by average wait time, number of customers, utilization, and probability of waiting. The Poisson distribution can model customer arrivals, and service times are often exponentially distributed.

1. WAITING LINES
DMA Kulasooriya
ISL Certified Six Sigma Black Belt
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

2. WAITING LINES
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

3. Waiting Lines
 Queuing theory: Mathematical approach to
the analysis of waiting lines.
 Goal of queuing analysis is to minimize the sum
of two costs
Customer waiting costs
Service capacity costs
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

4. Implications of Waiting Lines
 Cost to provide waiting space
 Loss of business
 Customers leaving
 Customers refusing to wait
 Loss of goodwill
 Reduction in customer satisfaction
 Congestion may disrupt other business
operations
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

5. Queuing Analysis
Optimum
Cost of
service
capacity
Cost of
customers
waiting
Total cost
Cost
Service capacity
Total
cost
Customer
waiting cost
Capacity
cost= +
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

6. System Characteristics
 Population Source
 Infinite source: customer arrivals are unrestricted
 Finite source: number of potential customers is limited
 Number of observers (channels)
 Arrival and service patterns
 Queue discipline (order of service)
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

7. Elements of Queuing System
Arrivals ServiceWaiting
line
Exit
Processing
order
System
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

8. Queuing Systems
Multiple channel
Multiple phase
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

9. System Performance
 Average number of customers waiting
 Average time customers wait
 System utilization
 Implied cost
 Probability that an arrival will have to wait
Measured by:
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

10. Waiting Time vs Utilization
System Utilization
Averagenumberon
timewaitinginline
0 100%
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

11. Queuing Models: Infinite-Source
 Single channel, exponential service time
 Single channel, constant service time
 Multiple channel, exponential service time
 Multiple priority service, exponential service
time
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

12. Queuing Models
 Single channel, exponential service time
 Single channel, constant service time
 Multiple channel, exponential service time
 Multiple priority service
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

13. Priority Model
Arrivals ServiceWaiting
line
Exit
Processing
order
System
11231
Arrivals are assigned
a priority as they arrive
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

14. Elements of Waiting line Analysis
A Queue: is a single waiting line
The Calling Population: is the source of customers : finite or
infinite
The Arrival Rate (λ): The frequency at which customers
arrive at a waiting line according to a probability distribution
For example if 100 customers arrive at a bank counter
during a 10-hour day, arrival rate average is 10 customers
per r hour. This is described by poisson distribution
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

15. Poisson Distribution
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5 6 7 8 9 10 11 12

16. The Poisson Distribution
Definitions
 An area of opportunity is a continuous
unit or interval of time, volume, or such
area in which more than one occurrence of
an event can occur.
 ex. The number of scratches in a car’s paint
 ex. The number of mosquito bites on a person
 ex. The number of computer crashes in a day
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

17. The Poisson Distribution
Properties
Apply the Poisson Distribution when:
 You wish to count the number of times an event occurs in a
given area of opportunity
 The probability that an event occurs in one area of
opportunity is the same for all areas of opportunity
 The number of events that occur in one area of opportunity
is independent of the number of events that occur in the
other areas of opportunity
 The probability that two or more events occur in an area of
opportunity approaches zero as the area of opportunity
becomes smaller
 The average number of events per unit is  (lambda)
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

18. The Poisson Distribution
Formula
X!
λe
P(X)
xλ

where:
X = the probability of X events in an area of opportunity
 = expected number of events
e = mathematical constant approximated by 2.71828…
6/28/2015DMA Kulasooriya, NIBM (UCD-BSc -2011)

19. The Poisson Distribution
Example
 Suppose that, on average, 5 cars enter a parking lot per
minute. What is the probability that in a given minute, 7
cars will enter?
 So, X = 7 and λ = 5
0.104
7!
5e
X!
λe
P(7)
75xλ


So, there is a 10.4% chance 7 cars will enter the parking in
a given minute.
DMA Kulasooriya, NIBM (UCD-BSc -2011)

20. Elements of Waiting line Analysis
Service Time: The time required to serve a
customer, is most frequently described by the
negative exponential distribution
The average service rate or how many customers
can be served in a period of time is expressed as μ
It is logical to assume that the rate at which services are
completed must exceed the arrival rate of consumers
λ < μ

21. Operating Characteristics
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 or being
served)
Wq – Average time a customer spends waiting in line
Po – Probability of no (zero) customers in the system
Pn – Probability of n customers in the system
P – Utilization rate : the proportion of time the system is in use

22. Probability of no customers in the system
Po = ( 1- λ/μ)

23. The Average Number of Customers in the
queuing system
L = λ/ μ-λ

24. Average Number customers in the waiting
line
Lq = λ2 /μ(μ-λ)

25. The average time a customer spends in the
system
W = 1/ μ –λ

26. The average time a customer spends
waiting in line to be served
Wq = λ/μ(μ-λ)

27. The probability that the server is busy and
a customer has to wait - utilization factor
P = λ/μ

28. The Probability that the server is idle and
a customer can be served
I = 1-p

29. A single Service Model
The fast shop drive in market has one check out counter where one
employee operates the cash register. The combination of the cash
register and the operator is the server ( service facility) in this
operating system. The customers who line up at the counter to pay
for their selections from the waiting line.
Customers arrive at a rate of 24 per hour according to the poisson
distribution. ( λ=24) and service times are exponentially
distributed with a mean rate of 30 customers per hour. (μ = 30)
The market manager wants to determine the operating
characteristics for this waiting line system

30. Probability of no customers in the system
Po = ( 1- λ/μ) = 1- 24/30
= 0.20
20% of the time there will be no
customers in the system

31. The Average Number of Customers in the
queuing system
L = λ/ μ-λ
L = 24/ 30- 24 = 4
4 customers on the average in the
queuing system

32. Average Number customers in the waiting
line
Lq = λ2 /μ(μ-λ) = (24)2 /30(30-24) = 3.2
The average customers in the
waiting line is 3.2

33. The average time a customer spends in the
system
W = 1/ μ –λ = 1/ 30-24 = 0.167 hour
( 10 minutes)
The Average time a customer spends in the
system is 10 minutes

34. The average time a customer spends
waiting in line to be served
Wq = λ/μ(μ-λ) = 24/30(30-24) = 0.133 hour
( 8 minutes)
Waiting time of a customer to be
served is 8 minutes

35. The probability that the server is busy and
a customer has to wait - utilization factor
P = λ/μ =24/30 = 0.80
80% of the time that the server is busy
and the customer must wait.

36. The Probability that the server is idle and
a customer can be served
I = 1-p = 1- 0.80 = 0.20
20% of the time a customer can be served
without waiting in the queue