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Unformatted text preview: Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahks material M/G/1 The service times are generally distributed X i : the service time of the ith arrival E[ X ] = 1/ = average service time E[ X 2 ] = second moment of service time Assume: Random variable X i s are identically and independently distributed Independent interarrival times PollaczeckKhinchin (PK) formula where W : the customer waiting time in queue W i : the ith customers waiting time in queue R i : the residual service time seen by the i th customer ) 1 ( 2 ] [ ) 1 ( 2 ] [ 2 2 2  = = X E N X E W Q Derivation of PK formula (since N i and X i are independent) ] [ ] [ ] [ ]]  [ [ ] [ ] [ 1 1 X E N E R E N X E E R E W E X R W i i i N i j i j i i i N i j j i i i i + = + = + =  = = W R W R N R W Q + = + = + = 1 Taking the limit, Define: M(t): the number of customers departing during [0, t ]  = 1 Then R W ] [ 2 1 ) ( lim ) ( lim 2 1 ) ( 1 lim 2 ) ( 1 2 X E t M X t t M d r t R t M t i t t t t = = = = The time average of the residual service time Take the limit as and assume the ergodic process ) ( ) ( 2 1 2 1 1 ) ( 1 ) ( 1 2 ) ( 1 2 t M X t t M X t d r t t M i i t M i i t = = = = t X Then Example: ) 1 ( 2 ] [ 2  = X E W ) 1 ( 2 ] [ 1 ] [ 2  + = + = X E W X E T 2 2 2 2 1 ] [ where M/D/1 for , ) 1 ( 2 2 ] [ where M/M/1 for , ) 1 ( = = = = X E X E W Example: Delay analysis of go back n ARQ...
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 Spring '08
 Choi

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