Stationary Point Processes, Palm Probabilities, and Queues
Ravi R. Mazumdar
Dept. of Electrical and Computer Engineering
University of Waterloo, Waterloo, ON, Canada
Workshop on Stochastic Processes in Engineering, IIT Bombay, March 14-15,
2013
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Overview of lectures
Lecture 1: Stationary point processes
Definition of point process.
Basic properties
Poisson process and Campbell’s Formula
Recurrence times and the inspection paradox
Stochastic intensity
Idea of Palm probability- conditioning at a pointPASTA
Basic formulae- Palm inversion, Papangelou’s formula: Sample-path
arguments
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Overview (contd)
Lecture 2: Palm theory and Rate Conservation Law
Lecture 3: More on queueing models
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Stationary Point Process
0
T1
T2 T3
T4
T5
T6
T7 t
Point process model
Let {Tn } denote a sequence of r.v.’s such that ...T−1 < T0 ≤ 0 < T1 <, · · · . The r.v’s correspond to a sequence of time
points. Assume that the sequence is stationary i.e Ti+1 − Ti are identically distributed.
Define:
X
Nt = N(0, t) =
1I[0,t) (Tk )
k
then Nt is said to be a simple point or counting process (it counts how many points lie in [0, t).
1
Nt counts the number of points Tn that occur in the interval (0, t]. The notation N(a, b] is also used to indicate the
number of points in (a, b]. Thus, Nt = N(0, t].
2
limt→∞ Nt = ∞ a.s. by definition.
3
The random variables Sn = Tn − Tn−1 are called the inter-arrival times.
4
The event {Nt = n} = {Tn ≤ t < Tn+1 }.
If {Sn } are identically distributed then Nt is called a stationary simple point process. If {Sn } are i.i.d then Nt is called
a renewal process. In particular if {Sn } are i.i.d. and exponentially distributed then Nt is called a Poisson process .
5
6
By definition of a simple point process, two events or arrivals cannot take place simultaneously (since
Tn < Tn+1 < Tn+2 · · · ).
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Poisson Process
Definition (Poisson Processes)
A renewal process {Nt }t≥0 is said to be a Poisson process with rate or intensity λ if the inter-arrival times {Sn } are i.i.d.
1.
exponential with mean λ
An immediate consequence (and often used as the definition without the need for the introduction of inter-arrival times) for
Poisson processes is the following that is more commonly found in texts:
Proposition (Equivalent Definition of Poisson Processes)
Let {Nt }t≥0 be a Poisson process with intensity λ. Then
1
N0 = 0
2
P(Nt − Ns = k) =
3
{Nt+s − Ns } is independent of Nu , u ≤ s.
(λ(t−s))k
k!
e −λ(t−s) for t > s.
An immediate consequence of the above definitions is that:
P(Nt+dt − Nt = 1)
=
λdt + o(dt)
P(Nt+dt − Nt ≥ 2)
=
o(dt)
Note the last property states that there cannot be more than one jump taking place in an infinitesimally small interval of time or
equivalently two or more arrivals cannot take place simultaneously.
A homogeneous Poisson process is a stationary independent increment procss
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Superposition, thinning, translations, and Poisson limits
Proposition
1
Let Nt1 and Nt2 be two independent Poisson processes with intensities λ1 and λ2 respectively. Then the process
Nt = Nt1 + Nt2 is Poisson with intensity λ1 + λ2 .
2
Suppose we randomly select points {Tn } of a Poisson process of intensity λ with probability p and define a new point
process N̂t as the resulting point process composed of the selected points. Then {N̂t } is Poisson with intensity λp.
3
Let {Tn } be the points of a Poisson process with intensity λ. Let {σn } be a collection of i.i.d. random variables, with
E[σi ] < ∞, and define:
0
Tn = Tn + σn
(1)
Let Nt0 be a point process whose points are {Tn0 }. Then Nt0 is a Poisson process with rate λ.
Theorem
(Poisson Limit)
Let {Nti }N
i=1 be a collection of independent stationary simple point processes with the following properties:
P
i
1 limN→∞ N
i=1 E[N1 ] = λ
PN
i
2
i=1 P(N1 ≥ 2) → 0 as N → ∞
Then
lim
N→∞
N
X
i weakly
Nt
−→
Nt
i=1
for every fixed t < ∞ where Nt is a Poisson r.v. with parameter λt.
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Non-homogeneous Poisson Process
Definition
{Nt }t≥0 is said to be a non-homogeneous Poisson process with intensity λt if:
1
N0 = 0
2
R
1 R t λ du k e − st λu du
For t > s, P(Nt − Ns = k) = k!
u
s
3
{Nt+s − Ns } is independent of Nu , u ≤ s.
Of course since λt depends on t we need to impose some conditions Ron the point process for it to be well-behaved, i.e. the
points do not accumulate. For this it is necessary and sufficient that 0∞ λs ds = ∞.
R
With this property it immediately follows that {limt→∞ Nt = ∞ a.s.} ⇐⇒ limt→∞ 0t λs ds = ∞ and
0 < T1 < T2 < · · · < Tn < Tn+1 < · · · and once again:
P(Nt+dt − Nt = 1)
=
λt dt + o(dt)
P(Nt+dt − Nt ≥ 2)
=
o(dt)
Note the last property states that there cannot be more than one jump taking place in an infinitesimally small interval of time or
two or more arrivals cannot take place simultaneously.
Definition
(Stieltjes Integral)
Rt
PNt
n=0 f (XTn ) can be written as 0 f (Xs )N(ds) where the integral is the so-called Stieltjes integral.
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Campbell’s Formula
We now see an important formula called Campbell’s Formula which is a smoothing formula.
Proposition
Let {Nt }t≥0 be a non-homogeneous Poisson process with intensity λt . Let {Xt } be a continuous-time stochastic process such
that Nt − Ns is independent of Xu , u ≤ s and let f (.) be a continuous and bounded function. Then:
Nt
X
E[
f (XT
Z t
n−
f (Xs )λs ds]
)] = E[
(2)
0
n=1
Proof It is instructive to see the proof. Let pn (t) be the probability density of the n-th point Tn ( which has a density given by
the Erlang-n distribution). Then from noting that the Poisson process has the density λt ,
pn (t) = λt
from which it readily follows that
Therefore
P∞
n=0
R
( 0t λs ds)n−1 − R t λs ds
0
e
(n − 1)!
pn (t) = λt .
Nt
X
E[
f (XTn − )]
Z t
∞
X
[E[
f (Xs− )|Tn = s]ds]
=
E
=
Z tX
Z t
∞
E[
E[f (Xs )λs ]ds
f (Xs− )pn (s)ds] =
0
n=1
n=1
0 n=1
0
where Xs− can be replaced by Xs because of continuity (in s).
Also since λt is non-random it can be removed from the expectation.
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Marked Poisson process and Poisson process in the plane
Consider the case of a so-called marked Poisson process which is the basic model that is used in the modeling of queues. This is
characterized by arrivals which take place as a Poisson process with rate or intensity say λ. Each arrival brings a mark
associated with it denoted by σn which are assumed to be i.i.d. In applications the mark denotes the amount of work or packet
length associated with an arriving packet. Then the process:
Xt =
X
n
σn 1I(0<Tn ≤t]
(3)
is called a Marked Poisson Process if the {Tn } correspond to the points of a Poisson process.
Suppose {σn } are i.i.d. with common probability density σ(x) then we can think of Xt as a two dimensional point process
whose points are (Tn , σn ) ∈ <2 .
Let Ai ⊂ <2 be defined as follows: Ai = {(x, y ) : x ∈ (ai , bi ], y ∈ (ci , di ]} and let:
X (Ai ) = {# of n : (Tn , σn ) ∈ Ai }
2
m
If {Ai }m
i=1 is collection of disjoint sets in < it is easy to see from the definition above that {X (Ai )}i=1 are independent and
moreover X (Ai ) is a Poisson r.v. with
µ(Ai )n −µ(A )
i
e
P(X (Ai ) = n) =
n!
Rd
where µ(Ai ) = λ|bi − ai | c i σ(x)dx by independence of {Tn } and {σn } and so in particular Xt can be seen as a
i
two-dimensional non-homogeneous Poisson process with intensity λσ(x).
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Martingales associated with point processes and stochastic
intensity
In the context
of point processes we will assume all the filtrations concerned are right continuous, i.e.
T
Ft+ = n≥0 F 1 = Ft . A process {Xt }t∈T is said to beFt predictable if Xt is Ft− measurable. So in particular Xt− is
t+
n
Ft predictable. Note Xt− is left-continuous by definition. Indeed, all left continuous Ft adapted processes are Ft predictable.
Predictable processes play a very important role in the stochastic calculus associated with point processes. We now briefly
indroduce the main results associated with stochastic intensities. Recall a stochastic process Xt is said to be a Ft - sub, super or
simply a martingale if for t ≥ s :
E[Xt |Fs ]
≥
Xs submartingale
=
Xs martingale
≤
Xs supermartingale
Note if Xt is a submartingale then −Xt is said to be a supermartingale.
Definition
Let (Ω, F , P) be a probability space on which a filtration Ft is defined. Let {Nt } be a simple point process adapted to Ft .
Then, there exists a non-negative predictable increasing process {At } called the compensator such that Nt − At is a Ft martingale.
If At is absolutely continuous with respect
to Lebesgue measure then ∃ a predictable intrgrable process λt ≥ 0 called the
R
stochastic intensity such that At = 0t λs ds.
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Thus the Poisson process corresponds to a point process whose stochastic intensity λ is non-random and a constant. Indeed it
follows that if a process has stationary independent increments then its intensity must be constant and therefore must be a
Poisson
R process (of course with respect to the given filtration). Note, a non-homogeneous Poisson process has a compensator
At = 0t λs ds and has independent but not stationary increments. Note by definition, every point process is a sub-martingale
with respect to the filtration on which it is defined.
The importance of predictability is the following. Let Mt be a right continuous Ft martingale and {Xt } be a Ft predictable
R
R
process with E[ 0T |Xt |2 dt < ∞. Then 0t Xs dMs is a Ft martingale.
To understand this issue of predictabilityRbetter let is consider
the following two integrals associated with a Poisson process with
R
intensity λ (viewed as Stiltjes integrals) 0t Ns dNs and 0t Ns− dNs .
The first integral is:
Z t
X
Nt (Nt + 1)
Ns dNs =
Ns (Ns − Ns− ) = 1 + 2 + 3 + · · · + Nt =
2
0
s≤t
and thus:
Z t
λ2 t 2
E[
Ns dNs ] =
+ λt
2
0
.
The second integral is
Z t
0
Ns− dNs =
X
Ns− (Ns − Ns− ) =
s≤t
X
Ns (Ns − Ns− ) −
s≤t
and thus
Z t
E[
0
Ns− dNs ] =
X
s≤t
2
(Ns − Ns− ) =
Nt (Nt − 1)
2
λ2 t 2
2
.
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On the other hand both:
Z t
Z t
Z t
λ2 t 2
2
E[
Ns λds] = E[
Ns− λds] =
λ sds =
2
0
0
0
Rt
Rt
and thus we see that taking Mt = Nt − λt: E[ 0 Ns dMs ] 6= 0 while E[ 0 Ns− dMs ] = 0 and thus only in the second case we
have:
Z t
Z s
Nu− dMu |Fs ] =
Nu− dMu
E[
0
0
R
showing that 0t Nu− dMu is a Ft martingale.
Clearly the results are different. In the first case the integrand is only adapted while in the second case it is predictable.
We can now reinterpret the previous results that we have seen before in the context of Campbell’s formula , i.e., (2). It is just a
re-statement of the martingale property, i.e., for any square-integrable Ft adapted process:
Z t
E[
0
Z t
f (Xs− )dNs ] = E[
Z t
f (Xs )λs ds] =
0
λs E[f (Xs )]ds
0
R
Here the martingale is Nt − At = Nt − 0t λs ds.
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Residual life and Inspection Paradox
Consider the following scenario: Let {Tn } be the points of a simple point process. Then the counting process, Nt , that we also
refer to as the point process is given by:
∞
X
Nt =
1I[0<Tn ≤t]
(4)
n=1
Now by definition TNt ≤ t < TNt +1 . We think of t ∈ [TNt , TNt +1 ) as an arbitrary observation point, i.e. if an observer
arrives at t then t must fall in the interval (TNt , TNt +1 ].
Proposition
Let {Ti } denote the points of a Poisson process with intensity or rate λ. Then conditionally on knowing that Nt = n, the
points {Tk }nk=1 are distributed as the order statistics of n- independent random variables that are uniformly distributed in [0, t].
Define the following:
R(t) = TNt +1 − t : the time the observer has to wait until next point after its arrival (at t). R(t) is called the forward
recurrence time or residual or remaining life (or time).
A(t) = t − TNt : referred to as the age or the last point before the observer’s arrival. A(t) is also called the backward
recurrence time .
Let S(t) = TNt +1 − TNt = A(t) + R(t) be the interarrival or inter-point time (seen by the arrival at t).
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Residual Life (contd)
1 . Thus one would expect
Now suppose Nt is Poisson with rate λ, then we know than Tn+1 − Tn is exponential with mean λ
1 . However this is not so! The fact that the observer arrives at t, biases its observation of the length of the
that: E[S(t)] = λ
interval- this is called the inspection paradox. Let us see why. ( Note TNt is random):
P(R(t) > x)
=
=
=
=
=
P(TNt +1 − t > x|TNt +1 > t)
P(TNt +1 > t + x|TNt +1 > t)
P(TNt +1 − TNt > t + x − TNt |TNt +1 − TNt > t − TNt )
Z t −λ(t+x−s)
e
e
e −λ(t−s)
0
−λx
dPT
Nt
(s)
R
where we used the fact that TNt +1 − TNt |TNt = s ∼ exp(λ) and 0t dPT (s) = 1 since TNt ≤ t by definition.
Nt
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Thus, what we have shown is that the residual life has the same distribution as the inter arrival times Tn+1 − Tn . Now since
S(t) = A(t) + R(t) and by definition A(t) ≥ 0 it means that:
E[S(t)] = E[A(t)] + E[R(t)] ≥
1
λ
so that the length of the interval in which the observer arrives is longer than the typical interval.
Let us see what the mean residual life is when the point process is not Poisson but a general renewal point process. Let
Sn = Tn − Tn−1 be the inter arrival time of arrivals from a general renewal point process and Sn are i.i.d. with distribution
R
Pn
1
F (t) with 0∞ x 2 dF (x) < ∞. Then Tn =
k=1 Sk . We assume E[Sn ] = λ ..
N
Now define Zi (t) = (Ti+1 − t)1I[T ,T
(t). Then:
i i+1 )
Z T
i+1
Ti
Define Z (t) =
P Nt
i=1
Zi (s)ds =
1
2
2
(Ti+1 − Ti ) =
Si2
2
Zi (t). Then by definition Z (t) = R(t):
1
Z t
t
0
Z (s)ds =
Z t
Nt
1 X
Si2
1
(TNt +1 − s)ds
+
t i=1 2
t TN
t
2
SN
R TN +1
Now second term on the r.h.s. is smaller than T t
Zi (s)ds = 2 t
Nt
Now letting t → ∞ we have:
P Nt
2
SN
S2
Nt
t
. k=1 k +
t
2(Nt )
2t
goes to λN 12 E[S 2 ] since
()
2
SN
t
t
goes to zero since S is asumed to have finite second moment.
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However,
lim
t→∞
1
Z t
t
0
Z (s)ds = E[R]
1 we have the main result for the mean residual time as:
and noting that λN = E[S]
E[R] =
1
1 E[S 2 ]
2 E[S]
1.
S is deterministic of length λ
Then, E[S 2 ] = 12 and
λ
E[R] =
2
=
1
2
E[S]
2 ].
S is uniform in [0, λ
We see that E[S 2 ] =
4
3λ2
and
E[R] =
3
1
2λ
2
3λ
=
2
3
E[S]
1.
S is exponential with mean λ
This is the Poisson case and gives, E[S 2 ] = 22 and
λ
E[R] =
1
λ
= E[S]
It can be seen that the higher the variability (as measured by the second moment) the higher the residual time, the deterministic
case being the smallest.
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Event and Time Averages: The appearance of Palm!
Let {Xt } be a stationary ergodic process with E[Xt ] < ∞. Then the ergodic theorem states that:
lim
t→∞
1
Z t
t
0
f (Xs )ds = E[f (X0 )]
for all bounded functions f (.). Now suppose we observe Xt at times {Ti } that correspond to a stationary and ergodic point
process Nt and we compute the average over the number of samples we obtain. Define the following if it exists:
d
EN [f (X0 )] =
lim
N→∞
N
1 X
N k=1
f (XT ) =
k
lim
t→∞
1
Z t
Nt
0
f (Xs )dNs
R
P Nt
f (XT ) by the definition of the Stieltjes integral . This is called an event average as
where we note that 0t f (Xs )dNs =
k=1
k
we have computed the limit over sums taken at discrete points corresponding to the arrivals of the point process. This arises in
applications when we can observe the contents of a queue at arrival times or at times when customers depart but not at all times.
A natural question is: Is EN [f (X0 )] = E[f (X0 )] ? The answer is no, in general. This is because the limit of the event averages is
computed under a probability distribution that corresponds to event or arrival times (that are special instants of time) that are
special. This is called a Palm probability or distribution. This differs from the time-stationary distribution (of the process {Xt }).
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Whetting our appetite: Palm probabilities and stationary
queueing systems
The idea of Palm probabilities is one of conditioning on a point in time where an event takes place.
Let {Nt } be a stationary point process and let {Xt } be a stochastic process defined on a probability space (Ω, Ft , P) on which
also {Nt } is defined.
A Palm probability tries to make sense of the following:
t
P (Xt ∈ A|∆Nt = 1)
i.e. the probability of Xt ∈ A when a point occurs.
Note the event ∆Nt = 1 occurs on a set of measure 0 and thus making sense of such a conditional probability needs some care.
Let us see some examples.
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Suppose {Nt }t≥0 is a Poisson process with intensity λ i.e. Nt − λt isa Ft martingale. Consider Xt too be Ft adapted.
E[Xt |∆Nt = 1]
=
=
lim
δ→0
lim
δ→0
E[Xt 1I[N[t,t+δ]=1] ]
E[1I(N[t,t+δ)=1]
E[Xt E[1I[N[t,t+δ)=1] |Ft ]
E[1I[N[t,t+δ=1] )|Ft ]
By definition of the stochastic intensity E[1I[N(t,t+δ)=1] |Ft ] = λδ + o(δ).
And hence we see that E[Xt |∆Nt = 1] = E[Xt ]
We can make this argument completely rigorous. The key is that conditioning w.r.t points of a Poisson process do not affect the
probabilities. This is an apparition of the so-called PASTA property. We will see this more in Lecture 2.
In general conditioning does affect the expectation.
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Let us see another example. Once again let {Nt } be a Poisson process with intensity λ.
Let {Xt } be a stochastic process adapted to Ft then
Z t
Z t
X
E[
Xs− dNs = E[
XTn − 1I[Tn ≤t] = λ
E[Xs ]ds
0
n
0
This is Campbell’s formula. In this case it just follows from the martingale property.
However when Nt is a stationary point process we can still obtain a similar formula if we replace the expectation on the r.h.s by
expectation w.r.t. Palm probability and λ by E[N[0, 1)]
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Let us now see a more general situation in the discrete-time case (when there is no problem in defining the conditional
probability). Let {ξn } be a a stationary sequence of {0, 1} r.v. with P(ξn = 1) = λ. LetP
{Tn } be the set of times when
n
ξn = 1 and we adopt the convention · · · < T−1 < T0 ≤ 0 < T1 < · · · . Let N(n) =
k=1 ξk .
We can now define for any A ∈ F :
n
P (Xn ∈ A)
=
=
=
P(Xn ∈ A|ξ(n) = 1)
P(Xn ∈ A, ξ(n) = 1)
P(ξ(n) = 1)
1
λ
P(Xn ∈ A, ξ(n) = 1)
The probability on the r.h.s is well defined since Nk , Xk are jointly defined.
Note by convention we take P 0 (T0 = 0) = 1
Now from above it follows:
X
E[
XTn 1I[0≤Tn ≤k] ]
=
n
k
X
E[Xn 1I[ξn =1] ]
n=0
=
(k + 1)λN EN [X0 ]
by the definition of PN (.) above. The only difference between this formula and Campbell’s formula (2) is that the underlying
point process was a continuous-time Poisson process in that context, and therefore instead of computing the mean with respect
to PN , the expectation there was computed with respect to the probability P, an issue that we will see a bit later in the context
of the PASTA property.
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In the continuous time case we can do the following: Let λ = E[N[0, 1)] Now clearly for any r.v. X we can define a measure
µ(.) for A ∈ < as follows:
X
1
µ(A) = E[X
1I[Tn ∈A] ]
λ
T
n
This is absolutely continuous
w.r.t. Lebesgue measure and hence by the Radon-Nikodym theorem we can define a density, say
R
p 0 (t). And µ(A) = A p 0 (s)ds where p 0 (t) = E[X |N({t} = 1]
Hence in particular:
Z
t
E[XN(A)] = λ
E [X ]dt
A
This is a special case of the Campbell-Mecke formula.
In lecture 3 we will see these concepts more rigorously.
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Inversion Formula and the Waiting Time Paradox
In general how do we relate the Palm probability and the reference probability?
This is given by the inversion formula:
0
T1 −1
E[Xk ] = E[X0 ] = λE [
X
Xk ]
k=0
Proof: First note by definition of the Palm probability above:
0
λE [
T1 −1
X
Xk ]
=
E[
T1 −1
X
k=0
Xk 1I[T =0] ]
0
k=0
=
E[
∞
X
k=0
=
E[X0
Xk 1Iξ =1;ξ ,ξ ,..,ξ
]
0
1 2
k−1 =0]
∞
X
k=0
where we have used stationarity in the last step. Now
P∞
k=0
1I[ξ
1I[ξ
−k =1;ξ−k+1 ,...ξ−1 =0]
−k =−1;ξm =0,−k+1≤m≤1]
]
= 1 a.s. since by definition
λ < ∞ and this just corresponds to stating that there exists a point before 0 at a finite distance. Hence the result follows.
1 obtained by taking X = 1.
A simple consequence of this result is E0 [T1 − T0 ] = λ
k
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In continuous-time the corresponding result is:
Z T
1
0
E[Xt ] = λE [
Xs ds]
0
where λ = E [N[0, 1)]
Let us see a sample-path proof under an ergodic hypothesis.
N
Proof: Let us assume both N and X are ergodic and hence: limt→∞ tt = λN a.s. by definition of λN as the average
intensity.
Without loss of generality let us take f (Xt ) = Xt . Then from the ergodicity of X. we have:
E[X0 ]
=
=
lim
t→∞
lim
t→∞
1
Z t
t
0
Xs ds
Nt
Nt 1 X
t Nt k=1
YT + lim
k
t→∞
1
t
Z t
Xs ds
TN
t
RT
= Tk
Xs ds and T0 = 0 by convention.
k−1
RT
P Nt
Y
goes to EN [ 0 1 Xs ds] by definition of YT and hence the result
Now as t → ∞ the last term goes to 0 while N1
k=1 Tk
k
t
follows.
where YT
k
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Some remarks are in order before we proceed.
Remark
By definition of PN we have PN (T0 = 0) = 1
From the inversion formula (by taking f (.) = 1) we see
EN [T1 ] = EN [Tn − Tn−1 ] =
1
λN
The interpretation of the expectation w.r.t. PN is that EN [f (X0 )] = E[f (Xt )|∆Nt = 1], i.e. we are conditioning at an
instant at which a point occurs.
From the definition of the Palm probability it can be seen that if {Xt } is a stationary process and Nt is a stationary
point process (consistent w.r.t. θt ) then:
Z t
E[
Xs dNs ] = λN tEN [X0 ]
0
Let us now see a consequence of the inversion formula: the famous inspection paradox.
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Let Nt be a point process.
Define: A(t) = TNt +1 − t. Then A(t) is the forward recurrence time- time to the next point given we arrive at t.
Similarly define B(t) = t − TNt the backward recurrence time. Then A(t) + B(t) = TNt +1 − TNt is the inter-point time
interval. By stationarity E0 [TNt +1 − TNt ] = E0 [T1 − T0 ] = E0 [T1 ] since under P 0 we have T0 = 0.
Taking Xt = T1 − t and Xt = t − T0 and using the inversion formula we have:
0
2
E[A(t) + B(t)] = E[T1 − T0 ] = λE [T1 ]
Noting λ = (E0 [T1 ])−1 and the fact that E[X 2 ] ≥ (E[X ])2 we see that E[T1 − T0 ] ≥ E0 [T1 − T0 ]. The exact difference is
var 0 (T1 )
.
E 0 [T1 ]
What this says is that observing an interval between two points biases us- i.e. if we arrive at arbitrary time between
two points, then we are more likely to arrive in a long interval.
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Two useful formulae
Proposition
Neveu’s Exchange Formula
Let N and N 0 be two stationary point processes and {Xt } be a stationary process all defined on (Ω, F , P) that are consistent
w.r.t. θt . Let λN and λN 0 be the average intensities of N and N 0 respectively and let {Tn } and {Tn0 } denote their points and
limn→∞ Tn = Tn0 = ∞ a.s.. Then for any measurable function such that EN [f (X0 )] < ∞:
Z T0
1
λN EN [f (X0 )] = λN 0 EN 0 [
f (Xs )dNs ]
(5)
0
where EN [.] and EN 0 [.] denote the expectations calculated w.r.t. the respective Palm probabilities.
The second formula of interest is one that allows us to relate computations between the Palm measure and the stationary
measure when we know the stochastic intensity of the underlying point process. As such it is a generalization of the EATA idea
when the underlying point process is not Poisson. In fact the importance of this result is that it ties in the Palm theory to the
martingale theory of point processes. This result is called Papangelou’s formula which is stated and proved exploiting the
martingale SLLN.
Proposition
Papangelou;s Formula
Let (Ω, F , P) be a probability space that carries a filtration Ft . Let {Xt } be a Ft -adapted stationary and ergodic process that
is jointly stationary with an ergodic point process {Nt } that possesses a Ft -stochastic intensity given by λt . Let PN and P
denote the Palm and stationary distributions for X. and λN = E[λ0 ] denote the average intensity.
Then:
λN EN [X0− ] = E[X0 λ0 ]
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Papangelou’s formula can be re-written as:
0
EN [f (X0− )] = E[f (X0 )] +
cov (f (X0 ), λ0 )
E[λ0 ]
where Xt is a stationary process and λt is the stochastic intensity of N.
To see how this result follows: Indeed for any X0 ∈ F0− , Papangelou’s formula reads as:
λN EN [f (X0 )] = E[f (X0 )λ0 ]
Now:
cov (f (X0 ), λ0 ) = E[f (X0 )λ0 ] − E[f (X0 )]E[λ0 ]
Noting that λN = E[λ0 ] we have:
EN [f (X0 )] =
cov (f (X0 ), λ0 )
E[λ0 ]
+ E[f (X0 )]
From here we immediately obtain the following result: if N is Poisson with (constant) intensity λ we see cov (f (X0 ), λ0 ) = 0
and thus:
EN [f (X0− )] = E[f (X0 )]
establishing PASTA. The above re-interpretation also shows that PASTA thus holds if λt and f (Xt ) are uncorrelated under P,
i.e. it can be extended to the doubly stochastic poisson case provided the intensity and the process {Xt } are uncorrelated as for
example in the situation when a continuous-time stationary stochastic process is sampled at times of a doubly stochastic Poisson
process independent of it.
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Motivation
Let {X (t)}, t ∈ R, be a real valued stochastic process and let N be a point process on R. The time average of {X (t)} up to
time t is
Z t
1
X (s)ds
Tt =
t 0
and the event average of {X (t)} up to time t is
Et =
1
N(0, t]
Z
X (s)N(ds)
(0,t]
The latter integral is interpreted as follows :
Z
X (s)N(ds) =
(0,t]
()
X
X (Tn )1[Tn ≤ t]
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When the processes are stationary and ergodic, (1.1) corresponds to the mean under the stationary measure while the event
average (1.2) converges to the mean under a measure termed the Palm probability.
The natural questions are how does one formally define the Palm probability and how does one compute it? What role does it
play in queues?
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Example: Jump distributions of Markov chains
Let {Tn } be points form a (strictly) increasing sequence of jump times of a Markov chain X (t)., with the property
limn→∞ Tn = +∞. Let Xn = X (Tn ) denote the discrete-time Markov chain viewed at the jump times.
Assume that {X (t)} is ergodic, with stationary distribution π, so that
lim
t→∞
1
Z t
t
0
1I[X (s)=i] ds = π(i)
In general, the imbedded Markov chain {Xn } is not ergodic, and when it is (under the condition that
qi−1
P
i∈E
π(i)qi < ∞ where
is the average sojourn time in state i between two jumps)
lim
t→∞
1
Z t
N((0, t])
0
1I[X (s)=i] ds =
lim
n
1 X
n→∞
n k=1
1I[Xn =i] = π0 (i)
where
π0 (i) = P
π(i)qi
j∈E
π(j)qj
6= π(i)
Equality π0 (i) = π(i) holds for all i ∈ E if and only if qi = constant, which is equivalent to {Tn } being a Poisson process.
What happens if x(t) is not Markov and the point process is not Poisson?
This will bring us to Palm probabilities.
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Palm Probability
Let (Ω, F , P) be a complete probability space which carries a measurable flow (shift) {θ}t . Let P be stationary w.r.t. {θt } i.e.
P ◦ θt
−1
=P
Let N be a point stationary point process (w.r.t the flow {θt } defined on (Ω, F , P)
N(θt ω, C ) = N(ω, C + t)
where C is a Borel set in <.
Let λN denote the average intensity of N given by:
λN = E [N(0, 1]]
The Palm Probability of (N,P) is defined by:
0
PN (A) =
1
λN `(C )
Z
E[
C
1A (θs )N(ds)
where `(c) denotes the Lebesgue measure of C and the definition does not depend on C .
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Properties of PN0
0
0
1. PN
(N{0} = 1) = PN
[T0 = 0] = 1
0
0
2. PN
◦ θTn = PN
0
.
3. EN
[T1 ] = λ−1
N
An immediate consequence of the definition is the so-called Mathes-Mecke formula
0
λN EN [
Z
<
Z
v (s)ds] = E[
<
v (0) ◦ θs N(ds)]
for any θt compatible process v (t)
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Stochastic Intensities and Martingale Representations
Let N be a simple, locally finite point process defined on R+ and let {Ft } be a
history of N satisfying the “usual” conditions. Then there exists a unique
nondecreasing process (unique within stochastic equivalence) {A(t)}, t ∈ R+ ,
that is Ft -predictable and right continuous such that A0 = 0 and
Z
Z
E[ X (s)N(ds)] = E[ X (s)A(ds)]
R
R
for all non-negative Ft predictable processes {X (t)}, t ∈ R. The process {A(t)}
is called the (P, Ft ) compensator of the point process N. Moreover ∆A(t) ≤ 1
and {ω |lim N(t, ω) = ∞} = {ω | lim A(t, ω) = ∞}. In fact, the process
{A(t)} is such that M = N − A is a local martingale .
If At is absolutely
R t continuous w.r.t Lebesgue measure, its density denoted by λt
given by At = 0 λs ds is called the Ft -(stochastic) intensity.
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If {X (t)} is any Ft predictable process such that for all t,
Z
|X (s)|A(ds) < ∞,
a.s.
(0,t]
the process {M̂(t)} defined by
4
Z
X (s) N(ds) − A(ds)
M̂(t) =
(0,t]
is a Ft -local martingale.
It is also true that the local martingale M = N − A is locally square integrable and more generally if for all t
Z
2
|X (s)| (1 − 4A(s))A(ds) < ∞,
a.s.
(0,t]
then M̂(t) is a local square integrable martingale. The condition above always for any bounded predictable process {X (t)} and
N locally finite.
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The quadratic variation process of M̂ denoted < M̂ > is defined as the unique predictable, non-negative and increasing process
that makes M̂ 2 − < M̂ > a local martingale. For the local martingale M̂ as defined earlier one has the explicit characterization
for the quadratic variation process
Z
2
< M̂ >t =
|X (s)| (1 − 4A(s))A(ds)
(0,t]
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Martingale SLLN
A. If Mt is a local square integrable martingale with a quadratic variation process < M > and if < M >∞ (ω) < ∞ then
Mt (ω)
B. If Mt is a local square integrable martingale with a quadratic variation process < M > and if < M >∞ (ω) = ∞ then
Mt (ω)
<M>t (ω)
→0
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Papangelou’s Theorem
One of the fundamental theorems that links the Palm probabilities with the
stochastic intensity theory is the Papangelou Theorem.
Theorem : PN0 << P on F0− iff N admits a Ft -intensity {λ(t)}. Moreover, in
that case λ(t, ω) = λ(0, θt ω) where
λ(0) = λN
dPN0 

dP F0−
Remark In particular Papangelou’s formula can be written as:
λN EN0 [X ] = E [λ(0)X ]
for all non-negative F0− measurable r.v. X .
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proof We will use ergodicity and the martingale SLLN to give a proof. For this let
us assume as in the proof of PASTA that supt E[|Xt |2 ] < ∞. Since by assumption
Rt
λt is the intensity and Nt is ergodic it follows that limt→∞ 1t 0 λs ds = λN a.s..
As in the proof of PASTA, using the SLLN as well as ergodicity, we can show that
Z
1 t
lim
Xs− (dNs − λs ds) = 0
t→∞ t 0
and hence once again re-writing the above we have:
( Rt
)
Z
Nt 0 Xs− dNs
1 t
lim
−
Xs λs ds = 0
t→∞
t
Nt
t 0
Noting by ergodicity limt→∞
result follows.
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Nt
t
= λN and the definition of Palm probabilities, the
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Next lecture:
Recapitulation
Rate Conservation Law (RCL) for stationary processes.
Swiss Army Formula (SAF) and applications
Queueing models: customer and work viewpoint.
Markovian queueing models introduction
Equilibrium distributions and Palm-Khinchine formula
Simple applications of RCL- Little’s formula, Pollaczek-Khinchine, etc.
Takac’s Equation
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Last lecture
The M/G /1 and G /M/1 queues
Busy period distributions
Insensitivity- M/G /∞ queue
Insensitive queueing models
Calculation of transient characteristics
Fluid Palm measures and fluid queues.
Further applications: Reflected diffusions with jumps
Skorokhod problem and queueing network models.
Chaoticity in queueing models.
Concluding remarks.
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References for Lecture 1
Palm Probabilities
F. Baccelli and P. Brémaud: Elements of Queueing Theory: Palm-martingale
and stochastic reurrences, 2nd. Edition, Applications of Math Series 26,
Springer-Verlag, NY., 2004.
K. Sigman: Stationary Marked Point Processes: An intuitive approach,
Chapman and Hall, N.Y., 1994
D. Daley and D. Vere-Jones; An introduction to the theory of point
processes, Springer-Verlag, 1988
R. R. Mazumdar, Performance modeling, stochastic networks, and statistical
multiplexing, Morgan and Claypool, Fall 2013 to appear. Revised version of
an earlier edition released in 2010.
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