Mahmoud Abas, Full-Stack Software Engineer & AI

Modeling and Simulation, Lab 1

Simulation Tables by Hand

Turn probabilities into random-number intervals, then build a simulation table one customer at a time.

Objectives

Turn a probability distribution into cumulative probabilities and random-number intervals
Read a random number into a value (an inter-arrival time, a service time, a category)
Build a simulation table for a single-server queue, column by column
Compute average waiting time, average time in system, and average idle time
Handle scheduled arrivals in clock time and problems with two independent random variables
Spot the classic mistakes: interval boundaries, max(), the first customer, units

From lecture 1

Where This Sits in the Course

Simulation: Imitating the operations of a facility or process, usually via computer.

System: A collection of entities (people, parts, machines, servers) that act and interact together toward some end.

Model: The assumptions, logical and mathematical, that we make to study the system.

Part 1

The Core Method

From a probability table to random-number intervals

The Four-Step Recipe

List every possible value with its probability
Add the probabilities to get the cumulative probability
Give each value a block of two-digit random numbers 00 to 99, as many numbers as its probability in percent
Read each given random number: find the interval it falls in, take that value

Worked Example: Q1 Service Times 1 to 4

Your turn: add up 0.10, 0.18, 0.22, 0.16, then turn each sum into an interval

Time	Probability	Cumulative	Interval
1	0.10	0.10	00 - 09
2	0.18	0.28	10 - 27
3	0.22	0.50	28 - 49
4	0.16	0.66	50 - 65

Worked Example: Service Times 5 to 8

Keep adding from 0.66: 0.14, 0.10, 0.05, 0.05

Time	Probability	Cumulative	Interval
5	0.14	0.80	66 - 79
6	0.10	0.90	80 - 89
7	0.05	0.95	90 - 94
8	0.05	1.00	95 - 99

The Interval Rule

L_k = C_k-1 × 100

H_k = C_k × 100 - 1

Service time 2: previous cumulative 0.10, this cumulative 0.28 gives 10 - 27
Service time 3: previous 0.28, this 0.50 gives 28 - 49
The last interval always ends at 99, a quick check that nothing is lost

Equally Likely Values

Q1 inter-arrival times 1 to 10: ten values share 100 numbers, 10 each

Inter-arrival	Interval	Inter-arrival	Interval
1	00 - 09	6	50 - 59
2	10 - 19	7	60 - 69
3	20 - 29	8	70 - 79
4	30 - 39	9	80 - 89
5	40 - 49	10	90 - 99

Quick Check: Read These Random Numbers

Q1 service-time table: 84, 27, 28, 50, 95

Random number	Interval	Service time
84	80 - 89	6
27	10 - 27	2
28	28 - 49	3
50	50 - 65	4
95	95 - 99	8

Part 2

The Simulation Table

One row per customer, one formula per column

Timing Columns: Arrival, Start, End

T_i = T_i-1 + A_i

B_i = max(T_i, E_i-1)

E_i = B_i + S_i

Result Columns: Wait, System, Idle

D_i = B_i - T_i

W_i = E_i - T_i

I_i = B_i - E_i-1

The Procedure for One Customer

Random number 1

interval lookup

Inter-arrival

Arrival time

Random number 2

interval lookup

Service time

Start

later of arrival and previous end

End

Wait, system, idle

Why max()? Two Cases

Column	Server already free	Server still busy
Arrival	after the previous end	before the previous end
Start	the arrival	the previous end
Wait	`0`	previous end minus arrival, above `0`
Idle	arrival minus previous end, above `0`	`0`

Part 3

Three Worked Problems

Q1 grocery store, Q2 dentist clinic, Q3 factory

Exercise 1

Q1: The Grocery Store

One counter; customers arrive 1 to 10 minutes apart, all equally likely
Service time 1 to 8 minutes with probabilities 0.10, 0.18, 0.22, 0.16, 0.14, 0.10, 0.05, 0.05
Random numbers for inter-arrival: 92, 73, 02, 95, 31, 92, 75, 24, 23, 30
Random numbers for service: 84, 10, 74, 53, 17, 79, 91, 67, 89, 38
Find arrival, service start and end, waiting time, time in system, idle time, and the averages

Q1 Step 1: Read the Random Numbers, Customers 1 to 5

Intervals from Part 1. RN 1: 92, 73, 02, 95, 31; RN 2: 84, 10, 74, 53, 17

Customer	Inter-arrival (RN 1)	Service (RN 2)
1	10 (92)	6 (84)
2	8 (73)	2 (10)
3	1 (02)	5 (74)
4	10 (95)	4 (53)
5	4 (31)	2 (17)

Q1 Step 1: Read the Random Numbers, Customers 6 to 10

RN 1: 92, 75, 24, 23, 30; RN 2: 79, 91, 67, 89, 38

Customer	Inter-arrival (RN 1)	Service (RN 2)
6	10 (92)	5 (79)
7	8 (75)	7 (91)
8	3 (24)	5 (67)
9	3 (23)	6 (89)
10	4 (30)	3 (38)

Q1: Customer 1

Inter-arrival 10, service 6, nobody ahead

Column	Working	Value
Arrival	0 + 10	10
Start	server free	10
End	10 + 6	16
Wait	10 - 10	0
System	16 - 10	6

Q1: Customer 2

Inter-arrival 8, service 2, previous end 16

Column	Working	Value
Arrival	10 + 8	18
Start	max(18, 16)	18
End	18 + 2	20
Wait	18 - 18	0
Idle	18 - 16	2

Q1: Customer 3

Inter-arrival 1, service 5, previous end 20

Column	Working	Value
Arrival	18 + 1	19
Start	max(19, 20)	20
End	20 + 5	25
Wait	20 - 19	1
Idle	20 - 20	0

Q1: Customers 4 to 7

Your turn: find each arrival, then compare it with the previous end

Customer	Arrival	Previous end	Idle
4	19 + 10 = 29	25	4
5	29 + 4 = 33	33	0
6	33 + 10 = 43	35	8
7	43 + 8 = 51	48	3

Q1: Customers 8 to 10, a Queue Builds

Short inter-arrival times 3, 3, 4 against service times 5, 6, 3

Customer	Arrival	Start (previous end)	Wait
8	54	58	4
9	57	63	6
10	61	69	8

Q1: The Result, Customers 1 to 5

Customer	Wait	Time in system	Idle
1	0	6	0
2	0	2	2
3	1	6	0
4	0	4	4
5	0	2	0

Q1: The Result, Customers 6 to 10

Customer	Wait	Time in system	Idle
6	0	5	8
7	0	7	3
8	4	9	0
9	6	12	0
10	8	11	0

Q1: The Averages

Predict first: out of 10 customers, how long does an average customer wait?

∑D_iN = 1910 = 1.9

∑W_iN = 6410 = 6.4

∑I_iN = 1710 = 1.7

Q1: Checking the Averages

Your turn: find the average service time, then add it to the average wait

From lecture 2

Extension: Server Utilization

ρ = ∑S_iE_N

Busy time: the sum of service times, 45 min
The last customer leaves at 72, so the run lasts 72 min
Idle over the whole run: 72 - 45 = 27 min

Q1: Server Utilization

Your turn: divide the busy time by the length of the run

ρ = 4572 = 0.625 = 62.5%

Exercise 1

Q2: The Dentist Clinic

6 patients by appointment: the first at 8:00, then one every 45 minutes

Service	Service time	Probability
Filling	45 min	0.40
Crown	60 min	0.15
Cleaning	20 min	0.15
Extraction	45 min	0.10
Checkup	15 min	0.20

Q2: Intervals, Then Treatments

Keep the table's order, Filling first. Cumulative: 0.40, 0.55, 0.70, 0.80, 1.00

Service	Interval	Random numbers	Patients
Filling	00 - 39	11, 34, 25	3, 4, 5
Crown	40 - 54	40	1
Cleaning	55 - 69	66	6
Extraction	70 - 79	none	none
Checkup	80 - 99	82	2

Q2: Working in Clock Time

Patient 1: crown, 60 min. Patient 2: checkup, 15 min

Column	Patient 1	Patient 2
Arrival	8:00	8:45
Start	8:00, dentist free	9:00, after patient 1
End	9:00	9:15
Wait	0	9:00 - 8:45 = 15

Q2: Patients 3 to 6

Your turn: compare each arrival with the previous end

Patient	Arrival and start	Service	End
3	9:30	45	10:15
4	10:15	45	11:00
5	11:00	45	11:45
6	11:45	20	12:05

Q2: The Average Wait

Your turn: add the waits of all the patients (6), then divide by 6

∑D_iN = 156 = 2.5

Exercise 1

Q3: Units and Vehicles

Find the average units produced and the average vehicles available per day, over 10 days

Units Xi	P(x = Xi)	Vehicles Yi	P(y = Yi)
500	0.05	5	0.15
550	0.15	6	0.35
600	0.25	7	0.20
650	0.35	8	0.18
700	0.20	9	0.12

Q3 Step 1: Both Interval Tables

Cumulative: units 0.05, 0.20, 0.45, 0.80, 1.00; vehicles 0.15, 0.50, 0.70, 0.88, 1.00

Units	Interval	Vehicles	Interval
500	00 - 04	5	00 - 14
550	05 - 19	6	15 - 49
600	20 - 44	7	50 - 69
650	45 - 79	8	70 - 87
700	80 - 99	9	88 - 99

Q3: Days 1 to 5

Units RN 91, 73, 02, 95, 31; vehicles RN 84, 10, 74, 53, 17

Day	Units (RN 1)	Vehicles (RN 2)
1	700 (91)	8 (84)
2	650 (73)	5 (10)
3	500 (02)	8 (74)
4	700 (95)	7 (53)
5	600 (31)	6 (17)

Q3: Days 6 to 10

Units RN 93, 75, 24, 23, 30; vehicles RN 79, 91, 67, 89, 38

Day	Units (RN 1)	Vehicles (RN 2)
6	700 (93)	8 (79)
7	650 (75)	9 (91)
8	600 (24)	7 (67)
9	600 (23)	9 (89)
10	600 (30)	6 (38)

Q3: The Averages

∑X_iN = 630010 = 630

∑Y_iN = 7310 = 7.3

Part 4

Common Mistakes

Where marks are lost, and how to check yourself

Mistakes with Intervals

Boundary off by one: cumulative 0.28 ends at 27, not 28
00 and 99: 00 is the first number and 99 the last; there are 100 numbers, not 99
Wrong table: reading a service number from the inter-arrival table
Wrong order: building intervals in a different order from the problem's table

Mistakes in the Table

Forgetting max(): starting a customer at the arrival while the server is still busy
First customer: check the problem. In lecture 1 customer 1 arrives at 0 with no random number; in Q1 customer 1 arrives after the first inter-arrival time (10)
Idle before the first customer: Q1 records 0 for customer 1; say clearly if you count the gap from 0
Units and clock time: 8:45 + 1:00 = 9:45 (60 minutes added), and waiting times are in minutes, not clock times
Averages: divide by the number of customers (or days), not by the last clock time

A Self-Check Before You Submit

The last interval of every table ends at 99
In every row, time in system is waiting time plus service time
In every row, at least one of waiting and idle is 0
Arrival times never decrease; service starts never come before the previous end
Totals are written under the columns, averages use the right count

Part 5

Practice: Your Turn

About 30 minutes. Answers follow each problem.

About 15 minutes

Practice 1: A Copy Center

One copy machine; customers arrive 1 to 5 minutes apart, all equally likely
Service time 1 to 5 minutes with probabilities 0.10, 0.25, 0.30, 0.20, 0.15
RN for inter-arrival: 85, 19, 98, 44, 53, 79, 52, 91
RN for service: 64, 99, 70, 09, 60, 05, 56, 77
Simulate 8 customers (customer 1 arrives after the first inter-arrival time, as in Q1)
Find average waiting time, average time in system, average idle time

Practice 1: Answer, Intervals

Time	Inter-arrival interval	Service cumulative	Service interval
1	00 - 19	0.10	00 - 09
2	20 - 39	0.35	10 - 34
3	40 - 59	0.65	35 - 64
4	60 - 79	0.85	65 - 84
5	80 - 99	1.00	85 - 99

Practice 1: Answer, Customers 1 to 4

Customer	Arrival	Start	End
1	5	5	8
2	6	8	13
3	11	13	17
4	14	17	18

Practice 1: Answer, Customers 5 to 8

Customer	Arrival	Start	End
5	17	18	21
6	21	21	22
7	24	24	27
8	29	29	33

Practice 1: Answer, Averages

∑D_iN = 88 = 1

∑W_iN = 328 = 4

∑I_iN = 48 = 0.5

Practice 1: Answer, Checking the Averages

Your turn: find the average service time, then add it to the average wait

About 10 minutes

Practice 2: A Car Garage

6 cars by appointment: the first at 9:00, then one every 40 minutes

Service	Service time	Probability
Oil change	20 min	0.35
Tire change	30 min	0.30
Brake repair	50 min	0.20
Full service	70 min	0.15

Practice 2: Answer, Intervals

Service	Cumulative	Interval
Oil change	0.35	00 - 34
Tire change	0.65	35 - 64
Brake repair	0.85	65 - 84
Full service	1.00	85 - 99

Practice 2: Answer, Cars 1 to 3

Car	Service	Start	Idle before
1	Tire change, 30 min	9:00	0
2	Oil change, 20 min	9:40	10
3	Full service, 70 min	10:20	20

Practice 2: Answer, Cars 4 to 6

Car	Service	Start	Wait
4	Oil change, 20 min	11:30	30
5	Brake repair, 50 min	11:50	10
6	Full service, 70 min	12:40	20
Total			60

Practice 2: Answer, Average Wait

∑D_iN = 606 = 10

About 8 minutes

Practice 3: A Bakery

Find the average orders per day and the average drivers available per day, over 8 days

Orders Xi	P(x = Xi)	Drivers Yi	P(y = Yi)
20	0.10	2	0.25
30	0.40	3	0.45
40	0.30	4	0.20
50	0.20	5	0.10

Practice 3: Answer, Intervals

Orders	Interval	Drivers	Interval
20	00 - 09	2	00 - 24
30	10 - 49	3	25 - 69
40	50 - 79	4	70 - 89
50	80 - 99	5	90 - 99

Practice 3: Answer, Days 1 to 5

Day	Orders (RN 1)	Drivers (RN 2)
1	20 (09)	4 (70)
2	40 (55)	2 (24)
3	30 (10)	5 (95)
4	50 (89)	3 (25)
5	30 (49)	3 (69)

Practice 3: Answer, Days 6 to 8

Day	Orders (RN 1)	Drivers (RN 2)
6	50 (80)	5 (90)
7	40 (50)	2 (00)
8	30 (26)	4 (89)
Total	290	28

∑X_iN = 2908 = 36.25 ∑Y_iN = 288 = 3.5

Key Takeaways

Probabilities become cumulative probabilities, then random-number intervals ending at 99
Each random number is read into a value through its own table
T(i) = T(i-1) + A(i) and B(i) = max(T(i), E(i-1))
D(i) = B(i) - T(i), W(i) = E(i) - T(i), I(i) = B(i) - E(i-1)
Totals under the columns, then divide by the number of customers or days

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Mahmoud AbasSimulation Tables by Hand