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Unformatted text preview: 1.2 Basic Queueing Models usc.den.ee555200901silvester A. Basic Queueing Concepts We think of a stream of packets arriving to a multiplexer as the customers in a queueing system. x n is service time for customer n t n+1 is interarrival time for customer n (between customer n and customer n+1) w n is waiting time for customer n s n = w n +x n is total system time for customer n See diagram on next page. Messages at a rate Server at rate = C/L Users Messages at a rate Server at rate = C/L Users 1.2 Basic Queueing Models usc.den.ee555200901silvester c n c n+1 c n+2 c n+3 c n c n+1 c n+2 c n+3 Departures Arrivals Queue In Service x n t n+1 w n+2 x n+2 x n+1 s n+2 IDLE BUSY PERIOD c n c n+1 c n+2 c n+3 c n c n+1 c n+2 c n+3 Departures Arrivals Queue In Service x n t n+1 w n+2 x n+2 x n+1 s n+2 IDLE BUSY PERIOD We note that: 1 1 , max{ + + + = n n n n t x w w } which is useful for studying system behaviour since x and t are random variables that can be simulated to drive system evolution. 1.2 Basic Queueing Models usc.den.ee555200901silvester ) ( ) ( ) ( t n t a t n = which is the number in system at time t Consider the area: = t dt t n t A ) ( ) ( Then: ) ( / ) ( t N t t A = is the average number (packets) in system in (0, t ). We can compute this area another way: = = ) ( 1 ) ( t a n n s t A (assume for simplicity that t is in an idle period). a ( t )= number of arrivals in (0, t ) d ( t )= number of departures in (0, t ) n ( t ) s n a ( t )= number of arrivals in (0, t ) d ( t )= number of departures in (0, t ) n ( t ) s n 1.2 Basic Queueing Models usc.den.ee555200901silvester So, ( ) 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) a t n n n s A t a t N t s t T t t t t a t = = = = = = Arrival rate in (0, t ) * Average time in system for packets up to time t Dropping the time parameter t , we have: T N = which is known as Littles Result . This is a basic and extremely valuable result. It can be applied to almost any system in which the arrivals and departures occurs in single steps (i.e. one at a time). Paraphrasing: the average time spent in a system is equal to the average number in the system divided by the rate of customers passing through the system. The system can be quite general, the queue, the system as a whole, just the server, a whole network of queues, etc. 1.2 Basic Queueing Models usc.den.ee555200901silvester B. M/M/1 The Simplest Queueing Model A simplifying assumption is that this arrival process is Poisson (exponential inter arrival time, independence between arrivals). The arrival rate is usually represented as (packets per second) and the mean interarrival time is 1 = t . The mean packet duration is x secs. For a message length, L ; and a channel capacity, C : C L x = . Again a (very) simplifying assumption is that the message length distribution is exponential with parameter x 1 = ....
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This note was uploaded on 12/22/2010 for the course EE 555 taught by Professor Silvester during the Fall '08 term at USC.
 Fall '08
 Silvester

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