Lecture11_2up

# Lecture11_2up - ECE-547 Introduction to Computer...

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1 Purdue University ECE ECE- 547 547 Introduction to Computer Introduction to Computer Communication Networks Communication Networks Instructor: Instructor: Xiaojun Xiaojun Lin Lin Lecture 11 Lecture 11 Purdue University M/M/1 system M/M/1 system ¾ P n = lim t P n (t). ¾ P n (t+ Δ t) = f(P j (t)) Question: If at time t+ Δ t the queue is in state n, then what are the possible states the queue can be in at time t? Probability of n packets in system at time t

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2 Purdue University M/M/1 system M/M/1 system ¾ Let n 1 a) At time t, Queue is in state n+2 or n-2 with probability o( Δ T) b) Queue is in state n at time t a) No arrivals or departure in (t, t+ Δ T] b) 1 arrival and 1 departure in (t, t+ Δ T] c) Any other scenario is o( Δ T) c) Queue is in state n-1 at time t a) One arrival and no departure b) o( Δ T) d) Queue is in state n+1 at time t a) No arrival and 1 departure b) Other scenario is o( Δ T) Purdue University M/M/1 system M/M/1 system ¾ P n (t + Δ t) = o( Δ T) + P n (t)[ μΔ t λΔ t + (1- λΔ t)(1- μΔ t) +o( Δ t)] + P n-1 (t)[(1- μΔ t) λΔ t + o( Δ t)] + P n+1 (t)[( μΔ t)(1- λΔ t) + o( Δ t)] ¾ Simplifying the above and incorporating all ( Δ
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