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Lecture 2 3 4

# Lecture 2 3 4 - 1.2 Basic Queueing Models A Basic Queueing...

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1.2 Basic Queueing Models usc.den.ee555-200901-silvester 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 inter-arrival 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 λ Server at rate μ = C/L Users 1.2 Basic Queueing Models usc.den.ee555-200901-silvester 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 , 0 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.

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1.2 Basic Queueing Models usc.den.ee555-200901-silvester ) ( ) ( ) ( t n t a t n = which is the number in system at time t Consider the area: = t dt t n t A 0 ) ( ) ( 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 1.2 Basic Queueing Models usc.den.ee555-200901-silvester So, ( ) 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) a t n n n s A t a t N t s t T t t t t a t λ = = = = = = Arrival rate in (0, t ) * Average time in system for packets up to time t Dropping the time parameter t , we have: T N λ = which is known as Little’s 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.