STAT253/317 Lecture 25 Yibi Huang 10.5..

Lecture 25 of STAT253/317, taught by Dr. Yibi Huang, focused on deepening students’ understanding of advanced probability and statistical inference, connecting theoretical concepts with practical problem-solving techniques. The lecture opened with a review of foundational probability distributions, emphasizing how these distributions form the basis for many statistical models. In particular, Dr. Huang highlighted the Poisson, Binomial, and Normal distributions as core examples, revisiting their moment generating functions and the role these play in deriving distributional properties.

Building on this, the session moved toward conditional probability and expectations. A central theme was the idea of conditioning as a tool to simplify complexity. Students were reminded that many difficult probability problems become manageable once we condition on an appropriate random variable or event. Dr. Huang introduced formal properties such as the Law of Iterated Expectation and the Law of Total Variance, both of which extend the power of conditional reasoning. Worked examples illustrated how these laws can be applied to scenarios ranging from risk assessment in insurance portfolios to queueing systems in operations research.

The lecture then transitioned into renewal processes and their applications. Renewal theory provides a framework for analyzing systems where events occur repeatedly over time, such as machine failures, arrivals of customers, or lifetimes of components. Dr. Huang explained the definition of a renewal process, the concept of interarrival times, and the key renewal theorem. Students were guided through proofs and intuitive explanations, showing how long-run averages can be derived even when the underlying distributions are not exponential. Particular attention was given to distinguishing renewal processes from Poisson processes, emphasizing that the exponential interarrival time distribution is what gives the Poisson process its unique memoryless property.

An advanced topic discussed in this lecture was the superposition of renewal processes. Dr. Huang carefully noted that while the superposition of two Poisson processes remains Poisson, the same is not true in general for renewal processes. Instead, such superpositions fall into a broader class of stochastic models, sometimes called Markov-renewal processes. A mathematical expression was introduced for the cumulative distribution of the first inter-event time under superposition, with discussion on its derivation and interpretation. This part of the lecture demonstrated how probability theory often balances elegance with complexity, as generalizations require careful handling of assumptions.

The session also touched on practical connections. Students considered how renewal models and their superpositions can be used to describe real-world systems like network traffic, machine maintenance schedules, and reliability analysis. By grounding abstract mathematics in applied contexts, Dr. Hu

1. STAT253/317 Winter 2019 Lecture 18
Yibi Huang
Section 7.7 The Inspection Paradox
Chapter 8 Queueing Models
Lecture 18 - 1

2. Section 7.7 The Inspection Paradox
Given a renewal processfN(t);t0gwith interarrival times
fXi;i1g, the length of the current cycle,
X
N(t)+1=S
N(t)+1S
N(t)
tend to belongerthanXi, the length of an ordinary cycle.
Precisely speaking,X
N(t)+1is Xi, which
means
P(X
N(t)+1>x)P(Xi>x);for allx0:
Lecture 18 - 2

3. Heuristic Explanation of the Inspection Paradox
Suppose we pick a timetuniformly in the range [0;T], and then
select the cycle that containst.
IPossible cycles that can be selected:X1;X2; : : : ;X
N(T)+1
IThese cycles are not equally likely to be selected.
The longer the cycle, the greater the chance.
P(Xiis selected) =Xi=T;for 1iN(T)
ISo the expected length of the selected cycleX
N(t)+1is roughly
N(T)
X
i=1
Xi
Xi
T
=
P
N(T)
i=1
X
2
i
T
!
E[X
2
i
]
E[Xi]
E[Xi] asT! 1:
ILast time we have shown that ifFis non-lattice,
lim
t!1
E[Y(t)] = lim
t!1
E[A(t)] =
E[X
2
i
]
2E[Xi]
;
SinceX
N(t)+1=A(t) +Y(t), limt!1E[X
N(t)+1] =
E[X
2
i
]
E[Xi]
Lecture 18 - 3

4. Example: Waiting Time for Buses
IPassengers arrive at a bus station at Poisson rate
IBuses arrive one after another according to a renewal process
with interarrival timesXi,i1, independent of the arrival of
customers.
IIfXiis deterministic, always equals 10 mins, then on average
passengers has to wait 5 mins
IIfXiis random with mean 10 min, then a passenger arrives at
timethas to waitY(t) minutes. HereY(t) is the residual life
of the bus arrival process. We know that
E[Y(t)]!
E[X
2
i
]
2E[Xi]

E[Xi]
2
= 5 min:
Passengers on average have to weight more than half the
mean length of interarrival times of buses.
Lecture 18 - 4

5. Class Size in U of Chicago
University of Chicago is known for its small class size, but a
majority of students feel most classes they enroll are big.
Suppose U of Chicago have ve classes of size
10;10;10;10;100
respectively.
IMean size of the 5 classes: (10 + 10 + 10 + 10 + 100)=5 = 28:
IFrom students' point of view, only the 40 students in the rst
four classes feel they are in a small class, the 100 students in
the big class feel they are in a large class.
Average class size students feel
40 students
z}|{
10 + + 10 +
100 students
z }| {
100 +: : :+ 100
140
=
1040 + 100100
140
74:3:
Lecture 18 - 5

6. Proof of the Inspection Paradox
Fors>x,
P(X
N(t)+1>xjS
N(t)=ts;N(t) =i) = 1P(Xi>x)
Fors<x,
P(X
N(t)+1>xjS
N(t)=ts;N(t) =i)
=P(Xi+1>xjSi=ts)
=P(Xi+1>xjXi+1>s)
=
P(Xi+1>x;Xi+1>s)
P(Xi+1>s)
=
P(Xi+1>x)
P(Xi+1>s)
P(Xi+1>x) =P(Xi>x)
ThusP(X
N(t)+1>xjS
N(t)=ts;N(t) =i)P(Xi>x) for all
N(t) andS
N(t). The claim is validated
Lecture 18 - 6

7. Limiting Distribution ofX
N(t)+1
If the distributionFof the interarrival times is non-lattice, we can
use an alternating renewal process argument to determine
G(x) = lim
t!1
P(X
N(t)+1x):
We say the renewal process is ON at timetiX
N(t)+1x, and
OFF otherwise. Thus in theith cycle,
the length of ON time is
(
XiifXix;and
0 otherwise
and hence
G(x) = lim
t!1
P(X
N(t)+1x) =
E[On time in a cycle]
E[cycle time]
=
E[Xi1
fXixg]
E[Xi]
=
R
x
0
zf(z)dz

In factG(x) =
x(1F(x))

+Fe(x)<Fe(x):
Lecture 18 - 7

8. Chapter 8 Queueing Models
A queueing model consists \customers" arriving to receive some
service and then depart. The mechanisms involved are
Iinput mechanism: the arrival pattern of customers in time
Iqueueing mechanism: the number of servers, order of the
service
Iservice mechanism: the time to serve one or a batch of
customers
We consider queueing models that follow the most common rule of
service: rst come, rst served.
Lecture 18 - 8

9. Common Queueing Processes
It is often reasonable to assume
Ithe interarrival times of customers are i.i.d. (the arrival of
customers follows a renewal process),
Ithe service times for customers are i.i.d. and are independent
of the arrival of customers.
Notation:M= memoryless, or Markov,G= General
IM=M=1: Poisson arrival, service timeExp(), 1 server
= a birth and death process with birth ratesj, and
death ratesj
IM=M=1: Poisson arrival, service timeExp(),1servers
= a birth and death process with birth ratesj, and
death ratesjj
IM=M=k: Poisson arrival, service timeExp(),kservers
= a birth and death process with birth ratesj, and
death ratesjmin(j;k)
Lecture 18 - 9

10. Common Queueing Processes (Cont'd)
IM=G=1: Poisson arrival, General service timeG, 1 server
IM=G=1: Poisson arrival, General service timeG,1server
IM=G=k: Poisson arrival, General service timeG,kserver
IG=M=1: General interarrival time, service timeExp(), 1
server
IG=G=k: General interarrival timeF, General service time
G,kservers
I: : :
Lecture 18 - 10

11. Quantities of Interest for Queueing Models
Let
X(t) = number of customers in the system at timet
Q(t) = number of customers waitng in queue at timet
Assume thatfX(t);t0gandfQ(t);t0ghas a stationary
distribution.
IL= the average number of customers in the system
L= lim
t!1
R
t
0
X(t)dt
t
;
ILQ= the average number of customers waiting in queue (not
being served);
LQ= lim
t!1
R
t
0
Q(t)dt
t
;
IW= the average amount of time, including the time waiting
in queue and service time, a customer spends in the system;
IWQ= the average amount of time a customer spends waiting
in queue (not being served).
Lecture 18 - 11

12. Little's Formula
Let
N(t) = number of customers enter the system at or before timet:
We deneabe the arrival rate of entering customers,
a= lim
t!1
N(t)
t
Little's Formula:
L=aW
LQ=aWQ
Lecture 18 - 12