Chapter 11: Waiting Line Models. Instructor: Dr. Neha Mittal. Queuing theory is the knowledge dealing with waiting lines. Waiting Line Models consist of mathematical formulas and relationships that can be used to determine the operating characteristics for a waiting line.
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Instructor: Dr. Neha Mittal
Queuing theory is the knowledge dealing with waiting lines.
Waiting Line Models consist of mathematical formulas and relationships that can be used to determine the operating characteristics for a waiting line.
They help in making decisions on the amount of capacity to be provided to give the desired service.
Providing too much service involves excessive cost
Providing too little service causes poor customer satisfaction, idle employees, etc.
Achieves economic balance between cost of service and cost associated with waiting for that service.Queuing Theory/ Waiting Line Models
Distribution of Arrivals
In most cases, the arrival of customers into the system is a random event
It is modeled as a Poisson process
Unusual behaviors (balking) are modeled using simulation.
Distribution of Service Times
Service time is also a random variable, in most cases.
A distribution commonly used to describe it is the exponential distribution.
These are the rules that determine the order in which arrivals will be serviced
Most common queue discipline is first come, first served (FCFS)
Others are FCLS, LCFS, LCLS, etc.
Can you think of an example of last come, first served (LCFS) queue discipline?
Some other disciplines assign priorities to the waiting units and then serve the unit with the highest priority first
A three part code of the form A/B/k is used to describe various queuing systems.
A identifies the arrival distribution
B the service (departure) distribution
k the number of channels for the system
When the queue discipline is FCFS, analytical formulas have been derived for several different queuing models including the following:
M/M/1 (focus for class)
M/G/k with blocked customers cleared
Analytical formulas are not available for all possible queuing systems. In this event, insights may be gained through a simulation of the system
Single channel/ single server
Poisson arrival-rate distribution
Exponential service-time distribution
First come first serve queue discipline
Unlimited maximum queue length (i.e., no balking)
Infinite calling population
Probability of n customers in queuing system
average number of customers in queuing system
average number of customers in waiting line
( ) ( )( )
Pn= ∙ P0 = 1 –
μ – λ
μ (μ – λ)
Lq =Single-Server Model
average time customer spends waiting in line
probability that server is busy and a customer has to wait (utilization factor)
probability that server is idle and customer can be served
μ – λλ
W = =
μ (μ – λ)
I = 1 – ρ
= 1 – = P0
M/M/1 Queuing System
Joe Ferris is a stock trader on
the floor of the New York Stock
Exchange for the firm of Smith,
Jones, Johnson, and Thomas, Inc.
Stock transactions arrive at a mean
rate of 20 per hour. Each order received by Joe
requires an average of two minutes to process.
What is the probability that no orders are received within a 15-minute period?
P (x = 0) = (50e -5)/0! = e -5 = .0067
What is the probability that exactly 3 orders are received within a 15-minute period?
P (x = 3) = (53e -5)/3! = 125(.0067)/6 = 0.1396
What is the probability that more than 6 orders arrive within a 15-minute period?
P (x > 6) = 1 - P (x = 0) - P (x = 1) - P (x = 2)
- P (x = 3) - P (x = 4) - P (x = 5)
- P (x = 6)
= 1 - .762 = .238
What is the mean service rate per hour?
Since Joe Ferris can process an order in an average time of 2 minutes (= 2/60 hr.), then the mean service rate, µ, is µ = 1/(mean service time), or 60/2.
m = 30/hr.
What percentage of the orders will take less than one minute to process?
Since the units are expressed in hours,
P (T< 1 minute) = P (T< 1/60 hour).
Using the exponential distribution, P (T<t ) = 1 - e-µt.
Hence, P (T< 1/60) = 1 - e-30(1/60)
= 1 - .6065 = .3935 = 39.35%
What percentage of the orders will be processed in exactly 3 minutes?
Since the exponential distribution is a continuous distribution, the probability a service time exactly equals any specific value is 0.
What percentage of the orders will require more than 3 minutes to process?
The percentage of orders requiring more than 3 minutes to process is:
P (T > 3/60) = e-30(3/60) = e-1.5 = .2231 = 22.31%
What is the average time an order must wait from the time Joe receives the order until it is finished being processed (i.e. in the system, its turnaround time)?
This is an M/M/1 queue with = 20 per hour and = 30 per hour. The average time an order waits in the system is: W = 1/(µ - )
= 1/(30 - 20)
= 1/10 hour or 6 minutes
What is the average number of orders in the line that Joe has waiting to be processed?
Average number of orders waiting in the queue is:
Lq = 2/[µ(µ - )]
What percentage of the time is Joe processing orders?
The percentage of time Joe is processing orders is equivalent to the utilization factor, /. Thus, the percentage of time he is processing orders is:
/ = 20/30
= 2/3 or 66.67%
1. In a waiting line situation, arrivals occur at a rate of 2 per minute, and the service times average 12 seconds. Assume the Poisson and exponential distributions.
2. During summer weekdays, boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of 3 per hour. In a 2-hour period,
a. what is the probability that no boats arrive?
b. what is the probability that 2 boats arrive?
c. what is the probability that 8 boats arrive?
3. The time to process a registration at the Sea View Resort follows the exponential distribution and has a mean of 6 minutes.
a. What is the probability of a registration time shorter than 3 minutes?
b. What is the probability of a registration time shorter than 6 minutes?
c. What is the probability of a registration time between 3 and 6 minutes?
4. The Grand Movie Theater has one box office clerk. On average, each customer that comes to see a movie can be sold its ticket at the rate of 6 per minute. For the theater's normal offerings of older movies, customers arrive at the rate of 3 per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution.