1. Discrete/Stochastic Simulation
Using PROMODEL

2. Usages Business Process Re-engineering Manufacturing Process Design
Service Process Design
Operations
Supply chains
As a planning tool
As an innovation and improvement tool

3. Applications of Discrete Stochastic Simulation
Resource management systems
Pollution management systems
Urban and regional planning
Transportation systems
Health systems
Criminal justice systems
Industrial systems
Education systems
eCommerce systems

4. What Discrete Stochastic Simulation isn’t
Stocks, states, rates, flows, information
Continuously changing variables
Causal loop diagramming
stock-and-flow diagramming

5. What Discrete Stochastic Simulation is
Probabilistic occurrences
Activity completions
Processes
Precedence relationships
Probabilistic routing
Events, Entities and Attributes

6. An example—Southwest Airlines airline turn--ACTIVITIES
Disembark passengers
Cabin cleanup
Embark passengers
Unload baggage
Load Baggage
Refuel
Remove waste
Refurbish snacks and drinks

7. EVENTS for the airline gate turn
Arrival at gate
Beginning of unloading
Completion of passenger unloading
Beginning of cleanup
Ending of cleanup
Beginning of passenger loading
Ending of passenger loading
Beginning of baggage unloading
Ending of baggage unloading

8. ACTIVITIES & EVENTS Activities always have time duration.
That time duration is in general random
Events are instants in time
Activities are sometimes engaged in by entities
Activities, events and entities have attributes

9. Events, Entities and Attributes
Entities may be permanent or temporary
Customers, students, piece parts, messages, boxes, items,--TEMPORARY
Universities, cities, companies, facilities, servers, professors, service areas --PERMANENT
Both entities and events possess ATTRIBUTES
Attributes of a server entity—mean service time, std. dev. of service time, distribution type
attributes of an event--event type, no of entities assoc. with it

10. A typical service system scenario
In the early morning hours between 7 and 9 a.m., arrivals to convenience stores are larger than normal. If there are more than 6 people waiting in line, new arrivals will balk and go somewhere else. People arrive at the rate of 1 every second, but the time is exponentially distributed. Patrons shop for a time period that is uniformly distributed between 3 and 5 minutes.

11. Convenience Store Scenario, continued
It takes the checkout clerk an average of 43 sec to collect money from a customer and provide them with a receipt, but this time is normally distributed with a std dev. of 30 sec. . People will automatically enqueue themselves in front of the checkout stand.
The manager can hire a second clerk, who is less well paid but also much slower

12. Convenience Store Scenario, continued
The store manager is interested in
the average waiting time of his patrons in the queue
the average number of customers that balked

13. With one store clerk: average waiting time is 128 seconds
number of balked customers is 76 out of 1000 customers
Check out clerk is busy 86% of the time checking out customers

14. With two store clerks:
average waiting time is 42 secs for the first server
69 secs for the second
There are no balked customers
Servers are busy 70% and 40% of the time, respectively

15. How does randomness come into play?
Probabilistic activity durations
Probabilistic routing “decisions”
Probabilistic arrivals

16. How is randomness created within a digital computer?
Monte Carlo--the computer generation of random numbers
Sample the clock?--no--not replicable
maintain a huge file of random numbers--no
Takes up too much space in primary memory
On secondary storage, its too slow
(when deciding to fetch from disk as opposed to primary memory, the time required. is 500,000 times longer
Use an ALGORITHM? --YES, YES

17. Why use an algorithm? The sequence it generates will be deterministic
Doesn’t take up much space in primary storage
Takes up no space on secondary storage

18. An Algorithm for Generating Random Numbers
Must be fast (short and sweet)
Must be capable of generating numbers that have all of the characteristics of randomness, but in fact are deterministic
Multiplicative Congruence is one method

19. About Random Numbers Uniform on the interval zero to one
They are completely independent and therefore un-correlated
We represent them this way: U(0,1)

20. Multiplicative Congruence Algorithm
CI+1 = K*Ci
function random(float u, int I)
I = I * ;
if I<0 then
I = I ;
else
U = I * E-9;
return “and” end;

21. Notes
Generates a sequence on the entire interval of 32-bit integers--0 to
Maps these onto the real interval of 0 to 1
If the first multiplication causes integer overflow, the resultant number I will be negative--it is made positive by adding the largest 32-bit integer representable +1
The last multiplication is like dividing the number by the largest integer possible
1/ = x10 to the minus 9

22. You can easily generate random numbers in an EXCEL spreadsheet using the function RAND()

23. What about non-uniform random numbers?
Exponential
Normal
Gamma
Poisson
Lognormal
Rectangular
Triangular

24. ONE ANSWER: Use the inverse transformation method
Every non-uniform random variate has an associated cumulative distribution function F(x) whose values are contained within the interval 0 to 1 and whose values are uniformly distributed over this interval
If x is a non uniform random variate, y = F(x) is uniformly distributed over the interval 0 to 1.

25. Strategy
If the inverse of the cumulative distribution function F(x) exists so that x = F-1(y) can be determined, then
1) simply generate a random number uniformly distributed on the interval 0 to 1
2) call this number y and apply the inverse transformation F-1(y) to obtain a random number x with the appropriate distribution.

26. Exponential Random variates
1) generate a random number U using the program given above
2)then apply EXPRND = -XMEAN * ALOG(U)
(This is simply using the inverse distribution method)

27. When Analytic inverses of the cumulative distribution function are unavailable
You can use a table function
You can use specialized algorithms that have been developed by academics over thirty-five years of cumulative research

28. Outputs The animation reveals bottlenecks We are also interested in
idleness, productivity, cycle time (time in the system), wait time, blocked time
number of trips made in a given period of time,
system throughput within a given period of time
We can get this from the statistical reports provided after the simulation is finished

29. Discrete stochastic simulation time advance
Time is advanced from event to event.
The only time instants looked at are the event times
The events are stored chronologically in time in an event file known as an “event calendar”
Corresponding to each event type is an event subroutine
When an event occurs, its subroutine is called

30. Discrete-stochastic simulation as
A statistical experiment
Running times must be long enough to ensure sufficient samples are collected
Several runs are often averaged together
The starting random number seeds are changed and the model is rerun
The basic idea is to get the variance to converge to the actual real-world variance

31. Another scenario
A mufflers-shocks-brakes shop is turning away business. It is considering hiring another mechanic or adding another bay. It currently has four bays.

32. Another scenario
A shipping company has just picked up additional customers and needs to add capacity. At its loading warehouse, it has four loading docks. It also has 10 trucks. Trucks currently wait upon return for four hours before they can go out on another trip. Should the company add docks, remove trucks, or both.

33. Another scenario
A ski rental shop fits customers for boots and then skis. It currently has four people working in the boot area and four people working in the ski area. Lines are very long and waiting times unacceptable. Should the shop hire more help or just shift some of its existing help from skis to boots or vice versa.

34. GO BACK TO 7-11 STORE example
Consider a 7-11 store in which the 7-9 a.m. period is of interest. Management is considering hiring a second clerk. Patrons arrive at the rate of one every 45 secs. Arrivals are Exponentially distributed. Patrons shop for a period that is uniformly distributed between 3 and 5 minutes. The check out time for each customer is normal with a mean of 43 secs and a std. dev. of 30 secs. Customers who encounter a queue of six customers or more upon arrival will walk away without shopping.

35. What are the activities?
Arrivals
Shopping
Checkout
What about waiting in Queue?
This is not an activity
This is handled automatically by the simulation

36. What units on time? Secs or mins? Let’s go with SECONDS
We must be consistent!!!!

37. Must also identify
Locations—points assoc with the starting and stopping events of an activity
Path network—the network the entity travels
Resources—permanent entities that act on ordinary temporary entities
Processes—the activities

38. PROMODEL SELECT BACKGROUND--optional BUILD-->locations
BUILD-->entities
BUILD-->PATH NETWORK
BUILD-->resources
BUILD-->processes and routing
BUILD-->arrivals
RUN IT

39. Locations Places where an event of importance to the model occurs
Like an arrival
A beginning of customer checkout
An ending of customer checkout

40. Entities
These are the temporary items that pass through the model of the system
Chits
Mail pieces
Piece parts
Students
Cars
People

41. Path network
The network that will be followed by the entities and/or the resources

42. Resources
Mobile permanent entities that can move over a network

43. Processes
A process is required everywhere the entity undergoes an operation
An exit process is always required

44. Routing You must specify how the entities move through the model
Usually you inform PROMODEL what path network to use

45. Arrivals
The statistics of the arrival process for each entity type must be communicated to PROMODEL

46. Now let’s look at PROmodel

47. Exercise 1. (15 points) A local convenience store has a self-service island from which it dispenses gasoline. Two lines of cars may form on either side of the island. The island will accommodate no more than two cars being filled with gas on a single side. There is space for no more than three cars in each of the two queues of cars waiting for each of the two service areas.

48. Cars arrive at the rate of one every minute with a distribution that is exponential. Service times are normal with a mean of seven minutes and a standard deviation of two minutes. Cars will drive away if more than six cars total are either waiting or in service (regardless of the line they are in). Once cars have entered the store’s gasoline facility, they will en-queue themselves into the shortest queue. Formulate a model in BLOCKS to determine how many cars are turned away in a day. For BRANCH/ TRANSFERS, be sure to indicate the type, such as UNCONDITIONALLY to block 12. Assuming the store is open 24 hours, setup the model to determine how many cars are turned away in one 24-hour day.

49. Promodel
What do we need to know??

50. What the following are Locations Entities Path networks Resources
Processes
Arrivals

51. What has to be specified before resources can be specified
Path networks

52. In order to get more than one arrival, the freq must be set to
INFINITE

53. In order to double the capacity of the number of turning machines and machining centers you would
Go to locations and increase the capacity from 1 to two for both of these locations

54. Failed Arrivals means
Arrivals that could not even get their foot in the door, in this case because the pallet was full

55. The important measures for this model were…
Throughput for the product
Utilization of the resource
Utilization of the locations
Failed arrivals
Amount of blocked time there is

56. On the final you will be given a scenario like the ones above and asked to determine
Locations
Entities
Path networks—may be asked to draw these
Resources
Processes
Arrivals

57. You will also need to know
what events and activities are
How random numbers are generated
How random variates (non-uniform) are generated
What is meant by MONTE CARLO