Management
Session Operations 16: Simulation
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Class Objectives
Generate random numbers.
Develop confidence intervals. Simulate an M/M/1 queue.
Simulate a call center.
The idea is to first set up a simulation for a model with which we are familiar (the M/M/1 queue), and then set up a simulation for a model whose behavior cannot be predicted analytically.
This is TN17, pages 693-716 in your text.
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Generating exponential random numbers
Assume U has a uniform distribution, so P(Uu) = u. (There are many algorithms, and much previous work on how to generate "truly" uniform random numbers.) Suppose we would like to generate an observation of the random variable X, which has an exponential distribution.
P( X x) F ( x) 1 exp( x)
Generate samples of the random variable U, and apply the formula ln(1-U)/(-).
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Why did that work?
ln(1 U ) P u Pln(1 U ) u P(1 U exp( u )) P(U 1 exp( u )) So, the random variable ln(1-U)/(-) has an 1 exp( u )
exponential distribution.
F (u ) P( X u )
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Example
1 2 0.787458 0.432425
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4
0.380017
0.420585
Uniform random numbers generated in Excel using the command: =RAND().
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6 7 8 9
0.315984
0.145883 0.818392 0.860034 0.332883
Note: the average of the ten numbers is 0.532. As more uniform random numbers are generated, that will approach 0.5.
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10
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0.824931
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Example Cont.
Apply the formula ln(1-u)/(-1) to generate samples from an exponential distribution with mean 1 (=1).
1 2 3 4 5 6 7 8 9 10
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0.787458 0.432425 0.380017 0.420585 0.315984 0.145883 0.818392 0.860034 0.332883 0.824931
1.548617 0.566382 0.478063 0.545737 0.379775 0.157687 1.705904 1.966357 0.404789 1.742576
=ln(1-0.787458)/(-1) Note: the average of the 10 numbers in the 3rd column is 0.950. As more random numbers are generated, this will approach 1.
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Example Cont.
Apply the formula ln(1-u)/(-2) to generate samples from an exponential distribution with mean (=2).
1 2 3 4 5 6 7 8 9 10
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0.787458 0.432425 0.380017 0.420585 0.315984 0.145883 0.818392 0.860034 0.332883 0.824931
0.774308 0.283191 0.239032 0.272868 0.189887 0.078843 0.852952 0.983179 0.202395 0.871288
=ln(1-0.787458)/(-2)
Note: the average of the 10 numbers in the 3rd column is 0.475. As more random numbers are generated, this will approach 0.5.
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What is the general formula?
Suppose we would like to generate samples of a random variable X having cdf F.
Step 1: solve for the inverse cdf F-1, F(F-1(x))=x.
Note that for an exponential random variable, F(x) = 1-exp(x) and F-1(x)=ln(1-x)/(-) so that F(F-1(x))=x.
Step 2: for each sample ui of a uniform random variable, apply the formula F-1(ui).
The numbers F-1(ui) are samples from the distribution F.
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Why does that work?
P F (U ) u Pln(1 U ) u
1
P(1 U exp( u )) P(U 1 exp( u ))
So, the random variable F-1(U) distribution F.
1 exp( u ) F (u ) P( X u )
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How many observations do we need?
Develop a confidence interval.
x / z / 2
Sample mean
n
Standard deviation
Number of observations
P(N(0,1)z)=
P x z / 2 , x z / 2 n n
True mean
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CI for Earlier Example
10 observations from a uniform(0,1) distribution (n=10).
Uniform(0,1) has standard deviation =(1/12)1/2. z/2=1.96 because P(N(0,1)1.96)=0.025.
Sample mean x is 0.532
1 / 12 1 / 12 P 0.532 1.96 ,0.532 1.96 10 10 P 0.353,0.711 0.95
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What does the CI mean?
There is probability 0.95 that the true mean of the distribution (0.5) falls within the interval [0.353,0.711]. In other words, if we were to generate many samples of 10 observations of a uniform(0,1) random number, 95% of those samples would fall within the interval [0.353,0.711].
As we generate more samples, the length of the interval decreases. (Go to Excel spreadsheet.)
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Simulating an M/M/1 Queue
Wn = wait time of nth customer to arrive Sn = service time of nth customer to arrive
Un= interarrival time of nth customer to arrive
Wn nth arrival Un
Sn nth arrival enters service
(n+1)th arrival
Wn 1 max Wn S n U n 1 ,0
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nth arrival departs; (n+1)th arrival enters service
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Simulating an M/M/1 Queue
= 0.9 = 1.0
Wait time = /[(-)] = 0.9/0.1=9 (Go to spreadsheet.)
Collect 10 average wait time observations from the spreadsheet: 8.3983, 9.6484, 7.8178, 11.0411, 7.7070, 9.2002, 9.5440, 8.3376, 8.2017, 9.8458 Average wait is: 8.97419
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Simulating an M/M/1 Queue
= 0.9, = 1.0, Wait time = /[(-)] = 0.9/0.1=9
10 average wait time observations from the spreadsheet: 8.3983, 9.6484, 7.8178, 11.0411, 7.7070, 9.2002, 9.5440, 8.3376, 8.2017, 9.8458 Average wait is: 8.974 Standard deviation is: 1.061 CI is: [8.974-1.96*1.061/sqrt(10), 8.974+1.96*1.061/sqrt(10)] = [8.316,9.632] There is 0.95 probability that the true wait time mean is in the interval [8.316,9.632].
Note: CI formula assumes variance is known, so is not quite right.
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Simulating a M/D/1 Queue
= 0.9, = 1.0, Wait time = /[2(-)] = 0.9/0.2=4.5 (formula from page 299 in textbook)
10 average wait time observations from the spreadsheet: 5.0105, 4.1717, 4.1744, 4.2995, 3.8020, 4.2256, 4.3264, 5.0001, 4.7881, 3.8670
Average wait is: 4.367 Standard deviation is: 0.430 CI is: [4.367-1.96*0.43/sqrt(10), 4.367+1.96*0.43/sqrt(10)] [4.10, = 4.63] Note that the wait time decreases as variability decreases.
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More complex simulations
It is often the case that we do not have analytic formulae to tell us process measurements like average customer wait time.
This is when simulations are particularly useful.
We will specify a more complex model, that cannot be analyzed using mathematical formulae, and use simulation to analyze it.
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Call Center Example
There are 4 call types and 4 agent types (not all of whom can answer every call).
Calls arrive and are put on hold (wait in queue) if necessary for an agent. Callers hang-up if they are put on hold for too long. An agent answers his specialized call type if any calls of that type are on hold, and otherwise answers his an optional call type.
If the queue is full, the call is blocked, and the caller receives a busy signal.
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Call Center Example: Agent-Call Matching
Call Type Agent Type Agent Type Agent Type Agent Type 1 2 3 4
1 2 3 4
X
O X
O 0 X X 0
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Call Center Example: Caller Statistics
Call Type Mean Inter-arrival Mean Renege Time (hours) Time (hours) Maximum Length
1 2 3 4
1.2 1.8 4.5 1.6
8 10 20 15
1000 1000 1000 1000
Inter-arrival and reneging times follow an exponential distribution.
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Call Center Example: Agent Statistics
Agent Type Number of Agents Mean Service Time (hours) Standard Dev. Service Time
1 2 3 4
7 8 8 12
10 15 30 12
2.5 2.5 2.5 2.5
Service times follow a lognormal distribution.
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Call Center Example: Questions of Interest
What is the average wait time? What is the utilization of each agent pool?
What percentage of calls are blocked or renege?
(Run Extend simulation, first with animation, then to 100,000 time units, without animation.)
(Note that this Extend demo can be downloaded from: http://www.extendsim.com/prods_demo.html)
ExtendSim OR
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Call Center Example: Questions of Interest
What is the average wait time?
0.4384 hours, or 26 minutes
What is the utilization of each agent pool?
Type 1: 0.816
Type 2: 0.847
Type 3: 0.907
Type 4: 0.767
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Call Center Example: Questions of Interest
What percentage of callers hang up without receiving service/ receive a busy signal?
Type 1 2 B 2061 1366 R 1690 437 Arrivals 81,135 54,550 %B 2.54% 2.50% %R 2.08% 0.80%
3 4
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526 1565
2587 1838
21,669 61,077
2.43% 2.56%
11.94% 3.01%
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Call Center Example: Conclusions
Too many type 3 callers are hanging up before reaching an agent.
Utilization of type 3 agents (which can only handle type 3 calls) was high (0.907). Maybe we need to hire more type 3 agents? Or, maybe we can train type 4 agents to also handle type 3 calls, since their utilization was lowest (0.767).
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Call Center Example: Conclusions
Decide on an appropriate strategy. Re-perform simulation, and see the effect of the strategy.
Compare the performance of different strategies.
Then, carry-forth the most cost effective strategy.
This is what happened in the INNOV8 game, at the boardroom, when you were asked about keeping calls in different queues, and outsourcing some of the calls.
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Summary: Major Phases in a Simulation Study
Define problem Construct simulation model Specify values of the variables and parameters
Run the simulation Evaluate results
Validation
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Propose new experiment
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Advantages of simulation
Developing the model of a system often leads to a better understanding of the real system. Time can be compressed in simulation; years of experience in the real system can be compressed into seconds or minutes. Simulation does not disrupt ongoing activities of the real system. Simulation is far more general than mathematical models, and can be used where conditions are not suitable for standard mathematical analysis.
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Advantages of simulation (cont.)
Simulation can be used as a game for training experience.
Simulation provides a more realistic replication of a system than mathematical analysis.
Simulation can be used to analyze transient conditions, whereas mathematical techniques usually cannot. Many standard packaged models, covering a wide range of topics, are available commercially.
Simulation answers what-if questions.
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Disadvantages of simulation
Although a great deal of time and effort may be spent to develop a model for a simulation, there is no guarantee that the model will, in fact, provide good answers.
There is no way to prove that a simulation model's performance is completely reliable. Simulation involves numerous repetitions of sequences that are based on randomly generated occurrences. An apparently stable system can, with the right combination of events however unlikely explode.
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Disadvantages of simulation (cont.)
Depending on the system to be simulated, building a simulation model can take anywhere from an hour to 100 worker years. Complicated systems can be very costly and take a long time.
Simulation may be less accurate than mathematical analysis because it is randomly based. If a given system can be represented by a mathematical model, that is better.
A significant amount of computer time may be needed to run complex models.
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