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Unformatted text preview: Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahks material Introduction Total delay of the ith customer in the system T i = W i + i N(t) : the number of customers in the system N q (t) : the number of customers in the queue N s (t) : the number of customers in the service W : the delay in the queue : the service time T : the total delay in the system : the customer arrival rate [#/sec] Littles Theorem E[N] = E[T] Number of customer in the system at t N(t)=A(t)D(t) where D(t) : the number of customer departures up to time t A(t) : the number of customer arrivals up to time t Time average of the number N(t) of customers in the system during the interval (0, t ], where N(t) = 0 = < t t dt t N t N ' ) ' ( 1 = = 1 1 t T i i A t ( ) < = t A t t ( ) < =< = N A t T t t i i A t 1 1 ( ) ( ) Let <T> t be the average of the times spent in the system by the first A(t) customers Then <N> t =<> t <T> t Assume an ergordic process and t , then E[N] = E[T] This relationship holds even in nonFIFO case < = = T A t T t i i A t 1 1 ( ) ( ) E[N q ] = E[W] Server utilization E[N s ] = E where utilization factor =/(c) < 1 to be stable for c server case Applications of Littles Theorem Example 3.1 : The average number of packets waiting in queue : The average time spent by a packet waiting in queue : The average transmission time : The arrival rate in a transmission line : The lines utilization factor Q N W = X = Q N W X Example 3.2 : The average delay : The average number of packets inside the network : The arrival packet rate of node i : The average number in the network of packets arriving at node i : The average delay of packets arriving at node i 1 n i i N T = = T N i i i i N T = i N i T Example 3.3: regular packet arrivals and departures : The packet arrival interval : The packet tx time ( ) : The processing and propagation delay per packet 1 K = K T K P = + P N T K = = + K P 1 < 2 K K P K < + < Example 3.5: Closed system with K servers and N customers : The number of servers : The average customer time in the system : The average customer service time N T = K X = N X T K = K T N K ( X If customers are blocked (and lost) if the system is full ( 29 1 K X = 1 K X =  1 K X  Since , K K ( The average number of busy servers: The blocking probability: Example 3.6: a roundrobin system for m users : The time interval used for overhead of user i : The average cycle length 1 2 m A A A A = + + + L 1 1 m i i i A X L = = + 1 1 m i i i A L X = = i A L Example 3.7 Combined with / N T = T R D = + R P T R NP + + N N R NP R P + + 1 P The # of terminals becomes the bottleneck The processing power becomes the bottleneck 1 min , N N R NP P R P & + + { } max , NP R P T R NP + + 1 / N R P < + 1 / N R P + Review of Markov chain theory...
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This note was uploaded on 04/29/2010 for the course CSE 4541.525A taught by Professor Choisunghyun during the Spring '09 term at Seoul National.
 Spring '09
 ChoiSunghyun

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