COMMERCE 3QA3 Lecture Notes - Tim Hortons, Standard Deviation, Auto Mechanic
4 Jun 2018
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Lectures 35-38 – Ch 9 (Queuing Models of Waiting Lines) page 1
Lectures 35-38 – Ch 9. Queuing Models of ‘Waiting Line Systems’
1. Introduction
The following elements comprise a waiting line system. Waiting line systems can be analyzed
using Simulation (ch. 10) or Queuing Models (ch. 9). Queuing models are easier.
Customer populations and service systems can be people waiting at a train station or check-in
counter, material waiting at a machine, packages and a pickup or delivery truck, machines
waiting for maintenance or repair, airplanes waiting to land or take off, etc.
The cause of waiting lines is variation from the random arrivals of customers and variation in the time
to serve customers. Each of these sources of variation is described with a probability distribution.
In ch. 9 we use the Poisson distribution for customer arrivals, and we use the exponential distribution
(and also a ‘deterministic distribution’ and a ‘general distribution’) for the time to serve customers.
1.1 Poisson distribution for customer arrivals (pp. 370-371) <click here to go to the podcast>
Customers arrive at the service system randomly. The probability that x = 0,1,2,3,4,5,...
customers arrive in a given time period is:
is the average arrival rate (i.e. is the average number of customers arriving in one time
period), e = 2.7183… (exponential constant)
The mean of this Poisson distribution is and the variance is .
Example
An average of 2 customers arrive each hour. What is the probability that x = 0,1,2,3,4,5,...
customers will arrive in one hour?
= average arrival rate = 2 customers/hour
So
,
,
, …
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Lectures 35-38 – Ch 9 (Queuing Models of Waiting Lines) page 2
Fig. 9.2 (p. 371)
The Poisson distribution is applicable when all of the following assumptions hold:
1. The average arrival rate in a given time period is known,
2. The average arrival rate is the same for all equal-sized time periods,
3. The actual number of arrivals in one time period is independent of the actual number of
arrivals in any other time period,
4. The number of arrivals cannot be more than one as the size of the time period approaches zero.
All the models in ch. 9 assume that customer arrivals follow a Poisson distribution. But other
distributions are also possible.
1.2 Exponential distribution for the service time (i.e. time to service a customer, pp. 372-374)
<click here to go to the podcast>
The time to serve one customer varies. The probability that the time to serve one customer is
less than or equal to t, and the probability that the time to serve one customer is greater than or
equal to t are: , and
µ is the average service rate (µ is the average number of customers completing service in one time
period), e = 2.7183… (exponential constant)
The mean of this exponential distribution is
and the variance is
.
Example
It takes an average of 20 minutes to serve a customer. What is the probability that a
customer will require 10 or less minutes of service; 10 or more minutes of service?
We can use any time period we like; suppose the time period is one minute. Then
average service time = 20 minutes per customer
µ = service rate =
= 0.05 customers per minute
t = 10 minutes
, and
‘rate’ means the time unit is in the denominator.
‘rate’ means ‘per hour’ (/hour).
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Lectures 35-38 – Ch 9 (Queuing Models of Waiting Lines) page 3
Suppose we set one time period equal to one hour. Then
average service time = 1/3 hour per customer
µ = service rate =
= 3 customers per hour
t = 1/6 hour = 0.1667 hours
, and
Fig. 9.4 (p. 374)
The exponential distribution is applicable when all of the following assumptions hold:
1. The average service rate µ in a given time period is known,
2. The average service rate µ is the same for all equal-sized time periods,
3. The actual number of customers completing service in one time period is independent of the
actual number of customers completing service in any other time period,
4. The number of customers completing service cannot be more than one as the size of the time
period approaches zero.
These assumptions correspond to the assumptions for the Poisson distribution.
Most of the queuing models in ch. 9 assume that service time follows an exponential distribution.
But ch. 9 also discusses queuing models where the service time is constant (i.e. deterministic)
and where the service time follows a general (or arbitrary) distribution.
Processes that follow a Poisson or an exponential distribution are called Markovian (M)
processes (named after the Russian mathematician Andrei Markov, 1856-1922).
Aside: The following is for information only. You are not responsible for it.
The Poisson distribution and the exponential distribution are closely related to each
other. If the number of arrivals follows a Poisson distribution then the time between
successive arrivals follows an exponential distribution.
‘rate’ means the time unit is in the denominator.
-‘per minute’ (µ = 0.05 customers/minute) is OK
-‘per hour’ (µ = 3 customers/hour) is OK
- µ = 20 minutes per customer is wrong
- µ = 1/3 hour per customer is wrong.
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