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Unformatted text preview: Basic Queueing Models EE555Spring 2008, Silvester 31 3 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. Basic Queueing Models EE555Spring 2008, Silvester 32 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. Basic Queueing Models EE555Spring 2008, Silvester 33 ) ( ) ( ) ( n a 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 Basic Queueing Models EE555Spring 2008, Silvester 34 So, ) ( ) ( ) ( ) ( 1 ) ( ) ( 1 t T t t a s t t a s t t N n t a n n λ = = = ∑ ∑ = = Arrival rate in (0, t ) * Average time in system for packets up to time t Dropping the time parameter...
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