Networking Theory and Fundamentals - Lecture 4 & 5 doc

4&5-2 TopicsMarkov Chains M/M/1 Queue Poisson Arrivals See Time Averages M/M/* Queues Introduction to Sojourn Times... 4&5-3 The M/M/1 QueueArrival process: Poisson with rate λ Service t

TCOM 501:

Networking Theory & Fundamentals

Lectures 4 & 5 February 5 and 12, 2003 Prof Yannis A Korilis

4&5-2 Topics

Markov Chains M/M/1 Queue Poisson Arrivals See Time Averages M/M/* Queues

Introduction to Sojourn Times

4&5-3 The M/M/1 Queue

Arrival process: Poisson with rate λ Service times: iid, exponential with parameter µ Service times and interarrival times: independent Single server

Infinite waiting room

N(t): Number of customers in system at time t (state)

4&5-4 Exponential Random Variables

4&5-5 M/M/1 Queue: Markov Chain Formulation

Jumps of {N(t): t ≥ 0} triggered by arrivals and departures

{N(t): t ≥ 0} can jump only between neighboring states

Assume process at time t is in state i: N(t) = i ≥ 1

X i: time until the next arrival – exponential with parameter λ

Y i: time until the next departure – exponential with parameter µ

T i =min{X i ,Y i }: time process spends at state i

T i : exponential with parameter νi= λ+µ

P i,i+1 =P{X i < Y i}= λ/(λ+µ), P i,i-1 =P{Y i < X i}= µ/(λ+µ)

P01=1, and T0 is exponential with parameter λ

{N(t): t ≥ 0} is a continuous-time Markov chain with

, 0 , 1

0, | | 1

i i i i i

i i i i i ij

4&5-6 M/M/1 Queue: Stationary Distribution

λ

µ

λ µ

4&5-7 The M/M/1 Queue

Average number of customers in system

Little’s Theorem: average time in system

Average waiting time and number of customers in the queue – excluding service

µ

λλ

λµ

W N

T

4&5-8 The M/M/1 Queue

0 2 4 6 8 10

ρ

ρ=λ/µ: utilization factor

Long term proportion of

time that server is busy

M/G/1 queue

Stability condition: ρ<1

Arrival rate should be less

than the service rate

4&5-9 M/M/1 Queue: Discrete-Time Approach

Focus on times 0, δ, 2δ,… (δ arbitrarily small)

Study discrete time process N k = N(δk)

Show that transition probabilities are

Discrete time Markov chain, omitting o(δ)

1 ( ), 1

( ), 0 ( ), 0 ( ), | | 1

ii

i i

i i ij

λδ λδ

4&5-10 M/M/1 Queue: Discrete-Time Approach

λδ µδ

λδ λδ

Discrete-time birth-death process → DBE:

Taking the limit δ→0:

4&5-11 Transition Probabilities?

A k : number of customers that arrive in I k =(kδ, (k+1)δ]

D k : number of customers that depart in I k =(kδ, (k+1)δ]

Transition probabilities P ij depend on conditional probabilities:

Calculate Q(a,d | n) using arrival and departure statistics Use Taylor expansion e -λδ=1-λδ+o(δ), e-µδ=1-µδ+o(δ), to express as

a function of δ

Poisson arrivals: P{A k ≥ 2}=o(δ)

Probability there are more than 1 arrivals in I k is o(δ)

Show: probability of more than one event (arrival or departure)

in I k is o(δ)

☺ See details in textbook

4&5-12 Example: Slowing Down

4&5-13 Example: Statistical MUX-ing vs TDM

m identical Poisson streams with rate λ/m; link with capacity 1;

packet lengths iid, exponential with mean 1/µ

Alternative: split the link to m channels with capacity 1/m each,

and dedicate one channel to each traffic stream

Delay in each “queue” becomes m times higher

Statistical multiplexing vs TDM or FDMWhen is TDM or FDM preferred over statistical multiplexing?

4&5-14 “PASTA” Theorem

Markov chain: “stationary” or “in steady-state:”

Process started at the stationary distribution, or

Process runs for an infinite time t→∞

Probability that at any time t, process is in state i is

equal to the stationary probability

Question: For an M/M/1 queue: given t is an arrival

time, what is the probability that N(t)=i?

Answer: Poisson Arrivals See Time Averages!

( )lim { ( ) } lim i

4&5-15 PASTA Theorem

Steady-state probabilities:

Steady-state probabilities upon arrival:

Lack of Anticipation Assumption (LAA): Future inter-arrival times and service times of previously arrived customers are independentTheorem: In a queueing system satisfying LAA:

1.If the arrival process is Poisson:

2.Poisson is the only process with this property (necessary and sufficient condition)

4&5-16 PASTA Theorem

Doesn’t PASTA apply for all arrival processes?

Deterministic arrivals every 10 secDeterministic service times 9 sec

Upon arrival: system is always empty a1=0

Average time with one customer in system: p1=0.9

“Customer” averages need not be time averagesRandomization does not help, unless Poisson!

1

4&5-17 PASTA Theorem: Proof

Define A(t,t+δ), the event that an arrival occurs in [t, t+ δ)Given that a customer arrives at t, probability of finding the system in state n:

A(t,t+δ) is independent of the state before time t, N(t-)

N(t - ) determined by arrival times <t, and corresponding service times A(t,t+δ) independent of arrivals <t [Poisson]

A(t,t+δ) independent of service times of customers arrived <t [LAA]

δ δ

4&5-18 PASTA Theorem: Intuitive Proof

ta and tr: randomly selected arrival and observation times, respectively

The arrival processes prior to ta and tr respectively are

The probability distributions of the time to the first arrival before ta

and tr are both exponentially distributed with parameter λ Extending this to the 2nd, 3rd, etc arrivals before ta and tr

establishes the result

State of the system at a given time t depends only on the arrivals (and associated service times) before t

Since the arrival processes before arrival times and random times are identical, so is the state of the system they see

4&5-19 Arrivals that Do not See Time-Averages

Example 1: Non-Poisson arrivals

IID inter-arrival times, uniformly distributed between in 2 and 4 secService times deterministic 1 sec

Upon arrival: system is always empty

λ=1/3, T=1 → N=T/λ=1/3 → p1=1/3

Example 2: LAA violated

Poisson arrivals

Service time of customer i: S i= αTi+1, α < 1

Upon arrival: system is always empty

Average time the system has 1 customer: p1= α

4&5-20 Distribution after Departure

Steady-state probabilities after departure:

Under very general assumptions:

N(t) changes in unit increments

limits a n and exist d n

a n = d n , n=0,1,…

In steady-state, system appears stochastically identical to an arriving and departing customer

Poisson arrivals + LAA: an arriving and a departing customer see

a system that is stochastically to the one seen by an observer looking at an arbitrary time

4&5-21 M/M/* Queues

Poisson arrival process

Interarrival times: iid, exponential

Service times: iid, exponential Service times and interarrival times: independent

N(t): Number of customers in system at time t (state)

{N(t): t ≥ 0} can be modeled as a continuous-time

Markov chain Transition rates depend on the characteristics of the system

PASTA Theorem always holds

4&5-22 M/M/1/K Queue

M/M/1 with finite waiting room

At most K customers in the system Customer that upon arrival finds K customers in system is dropped

1

n n

K

p

ρ ρ

1

K K

4&5-23 M/M/1/K Queue (proof)

ρ ρ

4&5-24 Truncating a Markov Chain

{X(t): t ≥ 0} continuous-time Markov chain with stationary distribution {p i : i=0,1,…}

S a subset of {0,1,…}: set of states; Observe process only in S

Eliminate all states not in S

Set

{Y(t): t ≥ 0}: resulting truncated process; If irreducible:

Continuous-time Markov chain Stationary distribution

Under certain conditions – need to verify depending on the system

if

0 if

j i

j i S

p

j S p

4&5-25 Truncating a Markov Chain (cont.)

Possible sufficient condition

Verify that distribution of truncated process

1. Satisfies the GBE

2. Satisfies the probability conservation law:

Another – even better – sufficient condition: DBE!

4&5-26 M/M/1 Queue with State-Dependent Rates

Arriving customer finds n customers in system

n < c: it is routed to any idle server

n ≥ c: it joins the waiting queue – all servers are busy

Birth-death process with state-dependent death rates

[Time spent at state n before jumping to n -1 is the minimum of

, 1 ,

n

µµ

4&5-29 M/M/c Queue

Probability of queueing – arriving customer finds all servers busy

Erlang-C Formula: used in telephony and circuit-switching

Call requests arrive with rate λ; holding time of a call exponential with mean 1/µ

c available circuits on a transmission line

A call that finds all c circuits busy, continuously attempts to find a

free circuit – “remains in queue”

M/M/c/c Queue: c-server loss system

A call that finds all c circuits busy is blocked

Erlang-B Formula: popular in telephony

4&5-30 M/M/c Queue

Expected number of customers waiting in queue – not in service

Average waiting time (in queue)

Average time in system (queued + serviced)

Expected number of customers in system

2

( ) ( ) ( ) ( )

! ! (1 ) (1 )

4&5-31 M/M/ ∞ Queue: Infinite-Server System

4&5-32 M/M/c/c Queue: c-Server Loss System

c servers, no waiting room

An arriving customer that finds all servers busy is blockedStationary distribution:

Probability of blocking (using PASTA):

Erlang-B Formula: used in telephony and circuit-switching

Results hold for an M/G/c/c queue

4&5-33 M/M/ ∞ and M/M/c/c Queues (proof)

4&5-34 Sum of IID Exponential RV’s

X1, X2,…, X n: iid, exponential with parameter λ

The probability density function of T is:

[Gamma distribution with parameters (n, λ)]

If X i is the time between arrivals i -1 and i of a certain type of events, then T is the time until the nth event occurs

For arbitrarily small δ:

Cummulative distribution function:

4&5-35 Sum of IID Exponential RV’s

Example 1: Poisson arrivals with rate λ

τ1: time until arrival of 1st customer

τi : ith interarrival time

τ1, τ2,…, τn: iid exponential with parameter λ

t n= τ1+ τ2+…,+τn : arrival time of customer n

t n follows Gamma with parameters (n, λ)

For arbitrarily small δ:

4&5-36 Sojourn Times in a M/M/1 Queue

Proof:

1. Direct calculation of probability distribution function

2. Moment generating functions

3. Intuitive: Exercise 3.11(b)

4&5-37 M/M/1 Queue: Sojourn Times (proof)

Proof 1: Let ti be the arrival time of customer i, and Ni = N (t¡i ), the number of customers in the system right before the ith arrival.

4&5-38 M/M/1 Queue: Sojourn Times (proof)

Proof 1: Note that:

² Time customer i stays in the system is greater than t, given that it ¯nds k customers in the system, i® the number of departures in interval (ti; ti + t) are less than k + 1 The server

is always busy during that interval, thus times between departures are iid, exponential with parameter ¹ Then:

4&5-39 M/M/1 Queue: Sojourn Times (proof)

Proof 2:

² N i : number of customers in system upon arrival of customer i

² T i(k): sojourn time of customer i when it ¯nds k customers in system

Ti(k) = S i + S i ¡1 + : : : + S i ¡k+1 + R i ¡k

S j is service time of customer j, and R i ¡k the residual service time

of the customer in service.

² S i ; : : : ; S i ¡k+1 : iid, exponential with parameter ¹

² R i ¡k : exponential with parameter ¹, independent of S i ; : : : ; S i ¡k+1

² T i(k) is the sum of k iid exponential RV's

² T i = T(Ni )

i is the sum of a random number of iid exponential RV's

² Use moment generating functions

4&5-40 M/M/1 Queue: Sojourn Times (proof)

4&5-41 Moment Generating Function

1 De¯nition: for any t 2 IR:

n

dt n M X (0) = E[Xn]

4 Moment Generating Functions and Independence:

X; Y : independent ) M X +Y (t) = M X (t)M Y (t) The opposite is nottrue.