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Unformatted text preview: Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material M/G/1 • The service times are generally distributed • X i : the service time of the ith arrival – E[ X ] = 1/μ = 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 customer’s waiting time in queue R i : the residual service time seen by the i th customer ) 1 ( 2 ] [ ) 1 ( 2 ] [ 2 2 2 ρ λ ρ λ = = 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 ρ μ λ μ + = + = + = 1 Taking the limit, Define: M(t): the number of customers departing during [0, t ] ρ = 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 λ τ τ = = = ∑ ∫ = ∞ → ∞ → ∞ → 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 ∑ ∑ ∫ = = = = τ τ t X Then Example: ) 1 ( 2 ] [ 2 ρ λ = X E W ) 1 ( 2 ] [ 1 ] [ 2 ρ λ μ + = + = 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 ( μ ρ μ ρ μ ρ μ ρ = = = = X E X E W • Example: Delay analysis of go back n ARQ...
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 Spring '08
 Choi
 Trigraph, Limit superior and limit inferior, Queueing theory, lim E Vl

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