Flows and Networks Plan for today (lecture 4):

Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Output simple queue • Tandem network • Jackson network: definition • Jackson network: equilibrium distribution • Partial balance • Kelly/Whittle network • Examples • Summary / Next • Exercises

Flows and NetworksPlan for today (lecture 3): • Last time / Questions? • Reversibility, stationarity • Truncation • Kolmogorov’s criteria • Poisson process • PASTA • Output simple queue • Tandem network • Summary / Next • Exercises

Poisson process • Definition : Poisson process :Let S1,S2,… be a sequence of independent exponential() r.v. Let Tn=S1+…+Sn, T0=0 and N(s)=max{n,Tn≤s}. The counting process {N(s),s≥0} is called Poisson process. • Theorem : If {N(s),s≥0} is a Poisson process, then(i) N(0)=0,(ii) N(t+s)-N(s)=Poisson( t), and(iii) N(t) has independent increments.Conversely, if (i), (ii), (iii) hold, then {N(s),s≥0} is a Poisson process

fraction of time system in staten probability outside observer seesncustomers at timet probability that arriving customer sees n customers at time t (just before arrival at time t there are n customers in the system) in general PASTA: Poisson Arrivals See Time Averages

Let C(t,t+h) event customer arrives in (t,t+h) For Poisson arrivals q(n,n+1)= so that Alternative explanation; PASTA holds in general! PASTA: Poisson Arrivals See Time Averages PASTA

Simple queue, Poisson() arrivals, exponential() service X(t) number of customers in M/M/1 queue: in equilibrium reversible Markov process. Forward process: upward jumps Poisson () Reversed process X(-t): upward jumps Poisson () = downward jump of forward process Downward jump process of X(t) Poisson () process Output simple queue

Let t0 fixed. Arrival process Poisson, thus arrival process after t0 independent of number in queue at t0. For reversed process X(-t): arrival process after –t0 independent of number in queue at –t0 Reversibility: joint distribution departure process up to t0 and number in queue at t0 for X(t) have same distribution as arrival process to X(-t) up to –t0 and number in queue at –t0. In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0 Holds for each reversible Markov process with Poisson arrivals as long as an arrival causes the process to change state Output simple queue (2)

Simple queue, Poisson() arrivals, exponential() service Equilibrium distribution Tandem of J M/M/1 queues, exp(i) service queue i Xi(t) number in queue i at time t Queue 1 in isolation: simple queue. Departure process queue 1 Poisson, thus queue 2 in isolation: simple queue State X1(t0) independent departure process prior to t0,but this determines (X2(t0),…, XJ(t0)), hence X1(t0) independent (X2(t0),…, XJ(t0)). Similar Xj(t0) independent (Xj+1(t0),…, XJ(t0)). Thus X1(t0), X2(t0),…, XJ(t0) mutually independent, and Tandem network of simple queues

Flows and NetworksPlan for today (lecture 5): • Last time / Questions? • Waiting time simple queue • Little • Sojourn time tandem network • Jackson network: mean sojourn time • Summary / Next • Exercises

Waiting time simple queue (1) • Consider simple queue FCFS discipline • W : waiting time typical customer in M/M/1(excludes service time) • N customers present upon arrival • Sr (residual) service time of customers present PASTA Voor j=0,1,2,…

Waiting time simple queue (2) • Thus • is exponential (-)

Little’s law (1) • Let • A(t) : number of arrivals entering in (0,t] • D(t) : number of departure from system (0,t] • X(t) : number of jobs in system at time t Equilibrium for t∞ In equilibrium: average number of arrivals per time unit = average number of departures per time unit

Little’s law (2) Fj sojourn time j-th departing job S(t) obtained sojourn times jobs in system at t

Little’s law (3) Assume following limits exist(ergodic theory, see SMOR) Then Little’s law

Little’s law (4) • Intuition • Suppose each job pays 1 euro per time unit in system • Count at arrival epoch of jobs: job pays at arrival for entire duration in system, i.e., pays EF • Total average amount paid per time unit  EF • Count as cumulative over time: system receives on average per time unit amount equal to average number in system • Amount received per time unit EX • Little’s law valid for general systems irrespective of order of service, service time distribution, arrival process, …

Sojourn time tandem simple queues Recall: In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0 Theorem 2.2: If service discipline at each queue in tandem of J simple queues is FCFS, then in equilibrium the waiting times of a customer at each of the J queues are independent Proof: Kelly p. 38 Tandem M/M/s queues: overtaking Distribution sojourn time: Ex 2.2.2

Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Output simple queue • Tandem network • Jackson network: definition • Jackson network: equilibrium distribution • Partial balance • Kelly/Whittle network • Summary / Next • Exercises

Simple queues, exponential service queue j, j=1,…,J statemovedepartarrive Transition rates Traffic equations Irreducible, unique solution, interpretation, exercise Jackson network: open Gordon Newell network: closed Jackson network : Definition

Simple queues, Transition rates Traffic equations Closed network Open network Global balance equations: Closed network: Open network: Jackson network : Equilibrium distribution

Transition rates Traffic equations Closed network Global balance equations: Theorem: The equilibrium distribution for the closed Jackson network containing N jobs is Proof closed network : equilibrium distribution

Global balance verified via partial balanceTheorem: If distribution satisfies partial balance, then it is the equilibrium distribution. Interpretation partial balance Partial balance

Transition rates Traffic equations Open network Global balance equations: Theorem: The equilibrium distribution for the open Jackson network containing N jobs is, provided αj<1, j=1,…,J, Proof Jackson network : Equilibrium distribution

Transition ratesfor some functions:S[0,), :S(0,) Traffic equations Open network Partial balance equations: Theorem: Assume that then satisfies partial balance, and is equilibrium distribution Kelly / Whittle network Kelly / Whittle network

Independent service, Poisson arrivals Alternative Examples

Simple queue s-server queue Infinite server queue Each station may have different service type Examples

Summary / next: Equilibrium distributions • Reversibility • Output reversible Markov process • Tandem network • Jackson network • Partial balance • Kelly-Whittle network NEXT: Sojourn times

Exercises [R+SN] 2.1.1, 2.1.2, 2.3.1, 2.3.4, 2.3.5, 2.3.6, 2.4.1, 2.4.2, 2.4.6, 2.4.7

Flows and Networks Plan for today (lecture 4):

Flows and Networks Plan for today (lecture 4):

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