132B_1_Sec11_DTQS_090710

# 132B_1_Sec11_DTQS_090710 - Discrete-Time Queueing Systems...

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Discrete-Time Queueing Systems Professor Izhak Rubin Electrical Engineering Department UCLA [email protected] © 2010-2011 by Izhak Rubin

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© Prof. Izhak Rubin 2 Geom/Geom/1 Queueing System z Single server queueing system z Time is slotted; slot duration = τ ; assume τ =1. z Arrivals and departures occur at slot start times. z Messages arrive in accordance with a Geometric arrival point process z M k = number of messages arriving in slot-k z {M k , k=1,2,3,…} = sequence of i.i.d. random variables z P{M k = 1} = p; P{M k = 0} = 1 - p; 0 < p 1. z t(j) = probability that time between two message arrivals (message inter-arrival time) is equal to j slots z t(j) = P(T n = j) = (1 – p) j-1 p; j = 1,2,3,…., 0 < p 1. z Message arrival rate = λ = 1 / E(T) = p [messages / slot]. Message arrivals Message departures Queueing facility, Buffer Service Channel; Server; e.g., processor; transmitter
© Prof. Izhak Rubin 3 Service Times The message service time is geometrically distributed S n = service time (measured in slots) required by the n-th message {S n , n=1,2,3,…} = sequence of i.i.d. random variables P{S n = j} = (1 – q) j-1 q; j = 1,2,3,…., 0 < q 1. E(S) = 1/q [slots/message]; μ = service rate = q [messages / slot, service-channel]

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© Prof. Izhak Rubin 4 Equivalent Service Model: Multi packet Messages Equivalently, the service time of a message is represented as follows: Assume the n-th message to consist of B n fixed size packets (or segments) The service time of a segment is equal to 1 slot . b(j) = probability that a message contains j packets = (1 – q) j-1 q; j = 1,2,3,…., 0 < q 1. b = average number of packets contains in a message = E(B) = 1/q [packets/mess]. Message departs system if its last packet departs (ends transmission) P(message in service departs at end of current slot) = P(current packet is last packet of this message) = q. Departure events of packets belonging to a message are statistically independent events due to the Geometric distribution of the service time.
© Prof. Izhak Rubin 5 System Size Processes z Due to the discrete time nature of the process, we set: z Arrivals that may occur during (k-1,k], the number of which is equal to N A (k)=M k , are said to occur at time k, and are recorded at time t = k +. z Departures that may occur at time k, the number of which is denoted as N D (k)=N Dk are recorded t = k - . z System size: X k = message system size at time k (t = k - ). This state is defined to not include arrivals that may occur during (k-1,k] [and be recorded at time t = k + ] but to include (i.e., account for) departures that may occur at time t = k - .

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© Prof. Izhak Rubin 6 System Size Processes (cont) z System size at time k+: X k + =X k+ ; X k + = X k + M k ; X k = X k-1 + -N Dk ; X k = X k-1 +M k-1 Dk z Packet Size (i.e., Packet system size): z X k
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132B_1_Sec11_DTQS_090710 - Discrete-Time Queueing Systems...

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