L. Kleinrock. Queueing Systems Vol I: Theory. John Wiley & Sons, New York, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... modeling [4] was pointed out [4] many e#orts have been devoted to study characteristics of Internet tra#c such as long range dependence, self similarity, and multi fractal scaling [6 8] The complexity of the packet arrival process has appeared to be intractable using traditional approaches [3], and it has even been suggested that an entirely new theory is necessary to develop tra#c engineering tools [5] To our knowledge, no study has yet attempted to predict the entire queue length distribution on an Internet link carrying realistic traffic such as that produced by finite TCP flows ....

L. Kleinrock, "Queueing Systems, Vol I: Theory," John Wiley and Sons, New York, NY, 1975.

.... the steadystate transition probabilities, i k j k Pr lim p s s k = 1 are obtained and the transition matrix P is formed [6] The steady state probability vector , whose elements are j is the solution to the finite set of linear equations = P, and , i i 1 [7], 8] Through computer simulations, we have obtained the behaviour of the steady state probabilities considering the 0 7803 7589 0 02 17.00 2002 IEEE PIMRC 2002 analysis made in [5] for a finite population and the limiting distribution [6] In Figure 1 we show, as an example, part of the ....

L. Kleinrock, Queueing Systems Vol I: Theory, ed. John Wiley & Sons, New York, 1975.

....[4] 11] where the term shot is synonymous here of flow rate function . In the particular case where 3 640 , that is, where shots are rectangles of height 1 and length , the process (1) is the number of clients found at time in an M G queue [17], if clients are identified with flows. We will allow however for shots with a more general shape than a rectangle of height 1, and we will see in this paper that this is indeed essential to characterize the total data rate on backbone links. We will use two alternative approaches to compute ....

....of the LST of and write it as a function of the random variables the rate function . We start by computing the PGF of , which is the number of clients found at time in an M G queuing model (Section IV) The distribution of in the stationary regime is given by (see e.g. 15] [17]) I Af( 40 103 hg f i . jIk ml Z . 0 is the load of the queue. As . 10 is finite, the system is stable and the number of active flows does not grow to infinity with probability 1. It follows that [1 n o g p J I 3 qAf )r[ 6CB D ....

L. Kleinrock, "Queueing Systems, Vol. I: Theory", Wiley, 1975.

....The fluid outflow represents the work performed per time unit by the server. In a single server system the work that is performed by the server is deterministic at rate one, as long as the queue is not empty. Hence, the flow rate r N(1; 0) The mean unfinished work in an M M 1 queue (see [10]) is U = 1 = 4:5; 20) for the values of the parameters chosen here. The fluid model is solved transiently. The solution is a full probability density from which the mean is computed. It is expected to approach the analytical solution as the model reaches equilibrium. In the M D 1 queue ....

....from which the mean is computed. It is expected to approach the analytical solution as the model reaches equilibrium. In the M D 1 queue a service time of 3=2 time units has been used. The analytical result for the stationary unfinished work is given by the Pollaczek Khinchin formula (see e.g. [10]) U = x 2 2(1 ) 2:25: 21) for the parameters chosen here. The curves for both service time distributions are shown in figure 2. Another model shall be regarded in order to study the usefulness of the fluid jumps. It is a performability model of a degradable buffered multiprocessor ....

L. Kleinrock. Queueing Systems Vol I: Theory. John Wiley, 1975.

....that assigns all message traffic to a single route will result in high queueing delays along that route. We model the stream of messages between a source and a destination as a stationary stochastic process. For example, this arrival process of messages may be taken to be a Poisson process [7]. Depending on the traffic routed over a given link, one can associate a similar stochastic process with the stream of messages arriving to that link from all contributing sources. Then, this arrival process together with the transmission characteristics of the link define a queueing process [7] ....

....[7] Depending on the traffic routed over a given link, one can associate a similar stochastic process with the stream of messages arriving to that link from all contributing sources. Then, this arrival process together with the transmission characteristics of the link define a queueing process [7] on that link. We consider the problem of minimizing some functional defined over the queueing processes of all links in the network, such as the mean packet delay for the network [5] In our formulation of single path routing, we do not focus on identifying the cost of a link in isolation but ....

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L.Kleinrock, Queueing Systems: Vol 1: Theory, Wiley, 1975.

....from a smaller state to a larger state (i.e. initial state has fewer voice slots allocated than the final state) y = 0 . The variable p i denotes the steady state probability of being in state i, and is determined by solving the equations p b i i M r d q = 1 0 and p p P tr = [9]. 3.2 Average Delay As mentioned in Section 2, voice calls are queued rather than blocked because small delays may be acceptable for voice traffic performance. This approach allows us to study the effects of data traffic on voice traffic delay, as well as determine how changing system parameters ....

....Viterbi define the voice Erlang capacity of a CDMA system in terms of the average number of users requesting service when the system maintains a blocking probability of 1 [10] Adequate voice service usually requires a blocking probability of less than 2 . Their model uses the Erlang B formula [9] which assumes newly arriving calls are lost if all slots are busy. While our model queues new arrivals rather than dropping them, we can still define acceptable system loads as those which keep the blocking probability ( 11 ) d f x n d i T x q = t t , 0 0 ( 12 ) E data ....

L. Kleinrock, Queueing Systems Vol 1: Theory, John Wiley & Sons, Inc., 1975.

....for x: x = L(x) r(t)e (1 x )t 0 dt (3.2) The solution of (3.2) can be obtained graphically, as shown in Fig. 3.1 0 x 1 1 u L(u) Fig. 3.1 Equation (3.2) has a unique solution in the interval [0,1) if l 1. Note that the equation (3. 2) is used in classical G M 1 queueing theory [7]. In the considered case l =0. Hence, the derivative of L(u) at u=1 is infinite. Let ti be the time of beginning the service of customer i. Define the i th customer delay by di = ti xi Denote d = lim di i Knowing qi , it is easy to find d : d = 1 iqi i =1 = x (1 x) ....

L.Kleinrock, "Queueing Systems-Vol. I: Theory", Wiley, N.Y., 1975

....method in [14, 30] and describe how derivative estimates may be obtained and used in Gallager s gradient based optimization algorithm. Let us define the following notation associated with a single queue in a queueing network [30] N j = the number of customers served in the j th busy period [17]. W ij = the waiting time of the i th customer in the j th busy period. W j = PN j i=1 W ij = the arrival rate to the queue. T j = the duration of the j th busy period. D = the expected steady state waiting time for the queue. We assume that the arrival process is Poisson. For a ....

.... which will be used to construct a more complex, derivative estimator: Y m 1 ( 1 m m X j=1 W j Y m 2 ( 1 m m X j=1 N j Gamma T j W j Y m 3 ( 1 m m X j=1 N j Gamma T j N j Y m 4 ( 1 m m X j=1 N j The steady state waiting time [17] can be written as a function of several variables identified with a regeneration period and when the likelihood ratio technique is applied to these relevant variables, we obtain the above formulas. The reader is referred to [30] for an intuitive explanation and thorough discussion of the these ....

L.Kleinrock, "Queueing Systems, Vol. I: Theory", Wiley, 1975.

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L. Kleinrock, Queueing Systems Vol 1: Theory, John Wiley & Sons, New York, NY, 1975. 23

....the system in a stable state. In a Round Robin processor sharing system, the average response time for requests requiring an average of x time units of processing is t = x 1 Gamma ae = s b(1 Gamma ae) 7) where ae is the system load, s is the average file size, and b is the system capacity[6, 7]. For the system shown in figure 3b, ae = s( 1 2 ) b. Therefore, in a multi user system, the cost of a normal request, which is the sum of the system resource cost and the delay cost, becomes c 1 = ff B Delta s ff T Delta t = ff B Delta s ff T Delta s b Gamma ( 1 2 )s (8) where b ....

L. Kleinrock, Queueing Systems Vol 1: Theory, John Wiley & Sons, New York, NY, 1975.

....the system in a stable state. In a Round Robin processor sharing system, the average response time for requests requiring an average of x time units of processing is t = x 1 Gamma ae = s b(1 Gamma ae) 7) where ae is the system load, s is the average file size, and b is the system capacity[6, 7]. For the system shown in figure 3b, ae = s( 1 2 ) b. This implies that, in a multi user system, the cost of a normal request, which is the sum of the system resource cost and the delay cost, becomes c 1 = ff B Delta s ff T Delta t = ff B Delta s ff T Delta s b Gamma ( 1 2 )s (8) ....

L. Kleinrock, Queueing Systems Vol 1: Theory, John Wiley & Sons, New York, NY, 1975.

....to keep the system in a stable state. In a Round Robin processor sharing system, the average response time for a request requiring x time units of processing is t = x 1 Gamma ae = s b(1 Gamma ae) 4) where ae = x is the system load, s is the average file size, and b is the system capacity [2,3]. Plugging (4) into (1) we obtain that, the cost of a normal request is c 1 = ff B Delta s ff T Delta s b Gamma ( 1 2 )s (5) where b ( 1 2 )s. Notice that in equation (5) the effect of prefetching to other users in the system is reflected through the 2 in the formula. As more ....

L. Kleinrock, Queueing Systems Vol 1: Theory, John Wiley & Sons, New York, NY, 1975.

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L. Kleinrock. Queueing Systems Vol I: Theory. John Wiley & Sons, New York, 1975.

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L. Kleinrock, "Queueing Systems, Vol. I: Theory", Wiley, 1975.

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L.Kleinrock, "Queueing Systems-Vol. I: Theory", Wiley, N.Y., 1975.

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