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# Metamaterials & Adv Ant Theory ECE 6962

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This 3 page Class Notes was uploaded by Shyanne Lubowitz on Monday October 26, 2015. The Class Notes belongs to ECE 6962 at University of Utah taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/230011/ece-6962-university-of-utah in ELECTRICAL AND COMPUTER ENGINEERING at University of Utah.

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Date Created: 10/26/15

ECE 6962003 queue size and queuing delay revisited 1 Introduction this document we review the behavior of an MMl queuing system as shown in Figure 1 In the MMl model one assumes that the packet arrivals into the queue as well as the departures out of the queue are random and occur with exponentially distributed interarrival timesl Therefore in any xed time interval the number of packets arrivals as well as packet departures is Poisson We further assume that ackets are serviced on a rstcome rstserve basis and that the system is in steady state We address queue sizes as well as queuing delays 1 v oru Figure 1 Queuing system with 3 packets in queue 2 Number of packets in the queue For the discussion on queuing systems the book uses the term quottmf c intensity I I 11L 1 R where L is the packet size R is the link speed and a is the offered load in packetssecond Note that is de ned to lie between 0 and 1 In the case of11 there are exactly as many packets arriving per unit of time say 1 second as can be serviced in that same unit of time say 1 second In other words since IaJR and according to the above de nition a is lower bounded by 0 and upper bounded by RL Clearly to know what values of a make sense for a given queuing system we need to know the packet size L and the link speed R The de nition of A is similar but also much more general at the same time Under the assumption of a certain arrival and service process the only relevant parameters to merely scalars that affect the shape of the delay curve not its geneml form Just like in the de nition of I we can now de ne an quotarrival mtequot Ian into the queue which lies between 0 and the rate out of the queue service rate Am Again we are not interested in In this notation the rst letter refers to the arrival process the second one to the departure process and the number equals the number of servers that service the queue Hence MMl stands for a Markovian input processMarkovian output process1 server queue Page 1 of3 ECE 6962003 queue size and queuing delay revisited values of km gt Am since they are guaranteed to cause in nite delay In the next step we can further simplify our de nition by expressing NH in terms of km such that Am lies in between Aquot LO and Aquot l 0M out On a side note whenever traf c is modeled with exponentially distributed interarrival times typically 9 not I is used for the performance evaluation of queuing systems and channel accessing techniques However you will nd all kinds of notations for example the utilization factor I as used in the hyperlink below Even if the variables have different names they all mean the same thing and lie between 0 and I From the derivation of the NUlVUl queue we are left with the nding that the average number of packets n in the queue follows from the average of the geometric distribution and can be expressed purely in terms of km according to aL A I 3 n m 77 2 1 tm 1 1 bag 0 You can nd the mathematical details that lead to this result at httpwwwmcsvuwacnzvignauddocsmmlhtmltth7sEc2 Note that I do not expect you to know the derivation In Figure 2 we compare simulation results of n to Eq 2 as a function of NH The following observations are made 39 The mean number of packets n in the system is greater than 0 even if the load measured by Mn is small 39 The mean number of packets n in the system tends to in nity as NH tends to 1 even though the service rate hm is still greater than the arrival rate kin quot theoretical value Figure 2 Simulation versus theoretical results of the queue size n as a function of km for an NUlVUl queuing system Page 2 of 3 ECE 6962003 queue size and queuing delay revisited 3 Queuing delay as a function of the offered load To nd the queuing delay that the packets have to face as a function of the offered load we can multiply n by the transmission speed LR out of the queue and the queuing delay dqueue follows immediately as d MEME 2 E q 1 Al R respectively aL LJE K E queue 1 I R 1 R R Page 3 of 3

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