1. Our favorite simple stochastic process.
QUEUING
Our favorite simple stochastic process.

2. SYSTEM DYNAMICS Body of Customers Server queue TIME = 0.0
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 0

3. I think I’ll get some service
SYSTEM DYNAMICS
Body of Customers
Server
queue
I think I’ll get some service
TIME = 0.0
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 0

4. SYSTEM DYNAMICS Body of Customers …this service will take 2.3 minutes…
Server
queue
TIME = 0.0
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 1

5. SYSTEM DYNAMICS …customer departs service Body of Customers Server
queue
TIME = 2.3
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 1

6. I think I’ll get some service
SYSTEM DYNAMICS
Body of Customers
Server
queue
I think I’ll get some service
TIME = 3.4
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 1
…this interarrival time is 3.4 minutes…

7. SYSTEM DYNAMICS …this service will take 5.3 minutes… Body of Customers
Server
queue
TIME = 3.4
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 2

8. I think I’ll get some service
SYSTEM DYNAMICS
Body of Customers
My departure time is 8.7
Server
queue
I think I’ll get some service
TIME = 4.6
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 2
…this interarrival time is 1.2 minutes…

9. SYSTEM DYNAMICS Body of Customers Server queue TIME = 4.6
TOTAL WAIT TIME = 0.0
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 3

10. I think I’ll get some service
SYSTEM DYNAMICS
Body of Customers
Server
queue
I think I’ll get some service
TIME = 4.9
TOTAL WAIT TIME = 0.3
TOTAL OBS. IN QUEUE = 0
ARRIVALS = 3
…this interarrival time is 0.3 minutes…

11. SYSTEM DYNAMICS Body of Customers Server queue TIME = 4.9
TOTAL WAIT TIME = 0.3
TOTAL OBS. IN QUEUE = 1
ARRIVALS = 4

12. SYSTEM DYNAMICS Body of Customers Server queue 0.3 + 2x(8.7-4.9)
TIME = 8.7
TOTAL WAIT TIME = 7.9
TOTAL OBS. IN QUEUE = 1
ARRIVALS = 4

13. COMMENTARY Body of Customers Server queue TIME = 8.7
The customer population is considered to be “large”
The summed of in-queue and in-service
(2 in this case)
is sufficient to predict the future
Body of Customers
Customers can recycle or go away
Server
queue
Our finite-duration simulation surrogates for an infinite length of time
TIME = 8.7
TOTAL WAIT TIME = 7.9
TOTAL OBS. IN QUEUE = 1
ARRIVALS = 4
Theses stats are used to calculate important measures of performance

14. SUMMARY STATS Body of Customers Server queue TIME = 10.0
TOTAL WAIT TIME = 7.9
TOTAL OBS. IN QUEUE = 1
ARRIVALS = 4
arrivals/min. = 4/10.0 = 0.4
wait/cust = 7.9/4 = 1.975
obs in queue = 0.25
These are samples (observations) from an unknown prob. distribution!

15. WHAT’S RANDOM? 2. Service times Body of Customers Server queue
1. Interarrival times
TIME = 8.7
TOTAL WAIT TIME = 7.9
TOTAL OBS. IN QUEUE = 1
ARRIVALS = 4

16. INDEPENDENCE Waiting times for these two customers are NOT independent
Body of Customers
Server
queue
TIME = 8.7
TOTAL WAIT TIME = 7.9
TOTAL OBS. IN QUEUE = 1
ARRIVALS = 4

17. Lindley’s Recursion Wait(8) Service(8) InArr(9) Wait(9) Service(9)
Wait(9) = Wait(8) + Service(8) – InArr(9)

18. DIVERSION: EXPONENTIAL RANDOM VARIABLES

19. Exponential Random Variables
DEFINITION
Let X be a random variable with distribution function
11/19/2018
Exponential Random Variables

20. Exponential Random Variables
f(x) is the DENSITY FUNCTION
F(x) is the DISTRIBUTION FUNCTION
Fc(x) is the SURVIVAL FUNCTION
l is the RATE, m is the EXPECTED VALUE
11/19/2018
Exponential Random Variables

21. Exponential Random Variables
DENSITY FUNCTION
area under the curve
is 1.0
11/19/2018
Exponential Random Variables

22. DERIVING EXPECTED VALUE
The Definition
of Expectation
11/19/2018
Exponential Random Variables

23. DERIVING EXPECTED VALUE
f(x) = 0 if
x < 0
11/19/2018
Exponential Random Variables

24. DERIVING EXPECTED VALUE
Integration by
parts
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Exponential Random Variables

25. DERIVING EXPECTED VALUE
Integral of the
density function
integrates to 1
“zero times
infinity” uses
L’Hopital’s
Rule
11/19/2018
Exponential Random Variables

26. DERIVING EXPECTED VALUE
Integration by parts
and Induction
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Exponential Random Variables

27. Exponential Random Variables
VARIANCE DERIVATION
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Exponential Random Variables

28. COEFFICIENT OF VARIATION
c.v. defined as the ratio of the mean to the standard deviation
standard deviation is SQRT(VAR(X))=1/l
c.v. for exponentials is always 1.0
11/19/2018
Exponential Random Variables

29. MIN OF TWO EXPONENTIALS
Let X1 and X2 be two exponential random variables
rates l1 and l2
independent
What’s the probability X1 is smaller than X2?
l1 / (l1+l2)
11/19/2018
Exponential Random Variables

30. DISTRIBUTION OF THE MINIMUM
Let Z = min(X1, X2)
then Z is exponentially distributed with rate l1+l2
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Exponential Random Variables

31. Exponential Random Variables
MEET THE SNAILS!
100cm 1
Snail covers 100cm in time X1
X1~expon(1.0 days)
E[X1] = 1/1
P[X1>1] = e-1=0.37
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Exponential Random Variables

32. Exponential Random Variables
100cm 1 1 1 1 n
Let Zn = winning time in an n-snail race
Zn ~ expon(nl)
E[Zn]=1/nl
lim E[Zn] = 0 as n gets large
Discuss common error of taking expectations too soon.
11/19/2018
Exponential Random Variables

33. RELEVANCE
N (large) customers independently decide when to go for service creating N go times
Customer #1’s go time is assumed to be 0.0
Customer #2’s inter-arrival time is the minimum of any remaining go times minus Customer #1’s go time
It is appropriate to model interarrival times as exponentials!

34. M/M/1 CONVENTION & NOTATION
l conventionally used as the parameter of the exponential distribution for arrivals
M (Markov) is a symbol for exponentially distributed inter-arrivals or service times
M/M/1
arrivals
departures
number of servers

35. LITTLE’S LAW Imagine that a customer pays $1/min. to stand in line
Let (0, T] be a long time interval
Let N(t) be the number of customers arriving in (0, T]
Let $ = proceeds in (0, T]

36. $1 = T * Avg system earnings per min
system earning per min = length of waiting line (L)
$2 = N(T) * Avg customer waiting cost
customer waiting cost = his waiting time (W)
$1 = $2

37. LITTLE’S LAW
TRUE FOR
ALL QUEUES!

38. EXTENSION
arrive ~ l
Service~m1
depart ~ l

39. EXTENSION m123 Service~m1 m2 m100 m4 m1 m666 arrive ~ l arrive ~ l
the
JACKSON NETWORK
arrive ~ l
arrive ~ l
m123
Service~m1 m2
probabilistic routing
is PP filtering
m100
depart ~ l m4 m1
m666