Lectures8_9 - M_G_1

# Lectures8_9 - M_G_1 - Lectures 8 9 M/G/1 Queues Eytan...

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M/G/1 Queues Eytan Modiano MIT Eytan Modiano Slide 1

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M/G/1 QUEUE Poisson Service times M/G/1 General independent Poisson arrivals at rate λ Service time has arbitrary distribution with given E[X] and E[X 2 ] Service times are independent and identically distributed (IID) Independent of arrival times E[service time] = 1/ µ Single Server queue Eytan Modiano Slide 2
Pollaczek-Khinchin (P-K) Formula W = λ E [ X 2 ] 2(1 − ρ ) where ρ = λ / µ = λ E[X] = line utilization From Little’s formula, N Q = λ W T = E[X] + W N = λ T= N Q + ρ Eytan Modiano Slide 3

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M/G/1 EXAMPLES Example 1: M/M/1 E[X] = 1/ µ ; E[X 2 ] = 2/ µ 2 W = λ µ 2 (1- ρ ) = ρ µ (1- ρ ) Example 2: M/D/1 (Constant service time 1/ µ ) E[X] = 1/ µ ; E[X 2 ] = 1/ µ 2 W = λ = ρ 2 µ 2 (1- ρ ) 2 µ (1- ρ ) Eytan Modiano Slide 4
Proof of Pollaczek-Khinchin Let W i = waiting time in queue of i th arrival R i = Residual service time seen by I (I.e., amount of time for current customer receiving service to be done) N i = Number of customers found in queue by i i arrives W i R i i-3 X i-2 X i-1 X X i Time -> N i = 3 i-1 W i = R i + X j j=i- N i E[W i ] = E[R i ] + E[X]E[N i ] = R + N Q / µ Here we have used PASTA property plus independent service time property

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Lectures8_9 - M_G_1 - Lectures 8 9 M/G/1 Queues Eytan...

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