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Example Send a file over the internet packet link (fixed rate) Modem card buffer

Delay Models place Computation transmission propagation (Queuing) C B time

Queuing Theory The theoretical study of waiting lines, expressed in mathematical terms input output server queue Delay= queue time +service time

Describe the dynamics of the system Evaluate its Performance The Problem Given One or more servers that render the service A (possibly infinite) pool of customers Some description of the arrival and service processes. Describe the dynamics of the system Evaluate its Performance If there is more than one queue for the server(s), there may also be some policy regarding queue changes for the customers.

(all the information necessary to completely describe the system) Common Assumptions The queue is FCFS (FIFO). We look at steady state : after the system has started up and things have settled down. State=a vector indicating the total # of customers in each queue at a particular time instant (all the information necessary to completely describe the system)

Notation for queuing systems omitted if infinite :Where A and B can be D for Deterministic distribution M for Markovian (exponential) distribution G for General (arbitrary) distribution

The M/M/1 System Poisson Process Exponential server output queue

Arrivals follow a Poisson process Readily amenable for analysis Reasonable for a wide variety of situations a(t) = # of arrivals in time interval [0,t]  = mean arrival rate t = k ; k = 0,1,…. ; 0 Pr(exactly 1 arrival in [t,t+]) =  Pr(no arrivals in [t,t+]) = 1- Pr(more than 1 arrival in [t,t+]) = 0 Pr(a(t) = n) = e- t ( t)n/n!

Model for Interarrivals and Service times Customers arrive at times t0 < t1 < .... - Poisson distributed The differences between consecutive arrivals are the interarrival times : n = tn - t n-1 n in Poisson process with mean arrival rate , are exponentially distributed, Pr(n  t) = 1 - e- t Service times are exponentially distributed, with mean service rate : Pr(Sn  s) = 1 - e-s

System Features  This is a Markovian System Service times are independent service times are independent of the arrivals Both inter-arrival and service times are memoryless Pr(Tn > t0+t | Tn> t0) = Pr(Tn  t) future events depend only on the present state  This is a Markovian System

Discrete Time Markov Nk =# at time k Pri,j=Pr(Nk+1= j | Nk= i) = probability to go from i to j events in the time interval  Pr0,0 1-   Pri,i+1    Pri,i-1    Pri,i  1- -   Pri,j  0 1-  -  1-   1-  -      1 2    

Steady State Solution # transitions kk+1 = # transitions k+1 k therefore: Prk ( )= Prk+1 ( ) Prk+1= (/) Prk = Prk  Prk= kPr0 and since: Prk=(1-) k

M/M/1 Performance Average # in the queue: Variance # in the queue

M/M/1 Performance is the mean waiting time to get through the queuing system Little’s Law : the mean # of customers in the system is 

A Heuristic Proof for Little’s law N(t)=A(t)-D(t)

Service Times Distributions for single server systems when service times (X1,X2,….) are i.i.d and independent of the interarrival times, the average service time: the expected customer waiting time in queue: (P-K formula) is the second moment for M/M/1 : for M/D/1 :

Service Times Distributions for single server systems (cont.) the total waiting time (in queue+in service): The expected # of customers in the system (Little’s law) : The expected # of customers in the queue

Networks of queues Interaction of communication lines transmission from one channel to another via junction useful application: estimate delay of packets in a packet switch. Problem: the interarrival times at the second queue depend on the packet lengths

Queuing networks : Model assumptions M/M/1 servers only ( , ) the output of each server is a Poisson similar to its input (Burke) the sum of several inputs I is a Poisson packets length is independent random (Kleinrock) 1 1+ 2+ 3 2 3