R. L. Cruz, "A calculus for network delay, Part II: Network analysis," IEEE Transactions on Information Theory, vol. 37, pp. 132 -- 141, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....VirtualClock scheme. This shows that the delay bound in Stop and Go can be much larger than the delay bound in the VirtualClock scheme. The reader is referred to [16] which compares the delay distribution seen by sessions under FCFS multiplexing with the delay bound computed with Cruz s method [2, 3], the delay bound using Stop and Go, and the delay bound using PGPS (presented later in this section) Hierarchical Round Robin (HRR) 10] also uses a framing strategy and is a non work conserving service discipline. It offers the same upper bound on delay as Stop and Go, but does not guarantee a ....

R. L. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," In IEEE Transactions on Information Theory, Vol. 37, No. 1, pp. 132-141, January 1991.

....INTRODUCTION Recent advances in the filtering theory under the (min; algebra (see e.g. 5] 1] 3] have shown that deterministic service guarantees can be achieved if traffic is properly constrained. The filtering theory is based on the traffic constraint functions (or sequences) in [6] [7]. For an arrival process A (with A(t) being the number of arrivals by time t) it is f upper constrained for some function f if A(t) Gamma A(s) f(t Gamma s) s t: 1) The tightest constraint function, known as the minimum envelope in [4] is A(t) sup s0 [A(t s) Gamma A(s) 2) To ....

R.L. Cruz, "A calculus for network delay, Part II: Network analysis," IEEE Transactions on Information Theory, Vol. 37, pp. 132-141, 1991.

....connections in multiplexing. The main objective of this paper is to evaluate performance bounds provided by the proposed scheme from an analytical viewpoint. Our analysis will concenlrate on the worst case performance of delay as well as jitter by assuming a leaky bucket constrained input source [13,14]. As will be shown in the following sections, the IRR server can effectively regulate the traffic flow of each connection within the network. This enables an internal node only to allocate a small buffer space for guaranteeing cell loss performance. In addition, the delay and jitter suffered by ....

....mer in the follong sections. 3 Characterization of Output Traffic The main difficulty of characterizing the traffic pattern for a connection inside the ATM network is the stochastic behavior becomes uncertain after being multiplexed with other connections. To deal with this problem, Cruz [13,14] proposed a (a, p) scheme for bounding the envelope of traffic pattern within the network, where p and a are the long term sustainable rate and the short term maximum burstiness, respectively. A cell stream is said to obey the (a, p) regularity, if the traffic amount generated during any time ....

R. Cruz, "A calculus for network delay, Part II: Network analysis," IEEE Trans. Inform. Theory, vol. 37, no. 1, pp. 132-141, Jan. 1991.

....# q where p and q are positive integers. Of course the delay can not decrease with more that the difference between the maximum and the minimum delay: ###### ; ## #### ### ### #n#### #n# # ## # (17) For an excellent and comprehensive treatise of this subject, see the work of Cruz [16] [17]. 4.4.2. Bounds of the variation of VBR queuing delays interface, then the delays encountered by the packets of any input source will have a delay variance bounded by: ## # # # # #n #### #n# # # # #: 18) Similar to the HFS case, the delay bounds in general are not integers and should be ....

R. L. Cruz, "A calculus for network delay, part II: Network analysis," IEEE Transactions on Information Theory, vol. 37, pp. 132--141, Jan. 1991.

....requirements. Generally, the end to end packet delay includes processing delay (packetizing, unpacketizing delay, etc. propagation delay, as well as queueing delay. Since the processing delay and propagation delay, which result from physical or technological constraints, are generally fixed [3], the design of a bounded delay service focuses on the study of queueing delay. Finding appropriate queue scheduling techniques has been considered as an important design aspect [4,5] Several scheduling techniques, such as the first come firstservice (FCFS) earliest deadline first (EDF) and ....

....techniques, such as the first come firstservice (FCFS) earliest deadline first (EDF) and staticpriority (SP) 1,2] have been studied. Each method presents a particular tradeoff in satisfying the requirements of efficiency, flexibility, complexity, analyzability, as well as impartiality [3]. The FCFS method, which is the simplest one, is very limited because it guarantees only one delay bound for all services in scheduling. The EDF method always selects the packet, which has the shortest deadline. Hence it can achieve the highest bandwidth utilization. Another advantage of the EDF ....

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R.A. Cruz, "Calculus for Network Delay, part II: Network Analysis," 1EEETram. on Comm., Vol. 37, No. I, pp. 13-147, January 1991.

....token bucket filter (rs, rsT) Thus, from (14) jI,N T N N aN dmax. s which is competitive with the s max,s s result of Stop and Go. The reader is referred to [25] which compares the delay distribution seen by sessions under FCFS multiplexing with the delay bound computed with Cruz s method [2, 3], the delay bound using Stop and Go, and the delay bound using PGPS (presented later in this section) Hierarchical Round Robin (HRR) 13] also uses a framing strategy and is a non work conserving service discipline. It offers the same upper bound on delay as Stop and Go, but does not guarantee a ....

....scheme because it closely emulates the service provided by a bit by bit round robin server. The reader is referred to [12] for a relevant work on fair queueing systems. Others have addressed the possibility of providing per session bounds on delay and delay distribution in a network setting [2, 3, 15, 24]. In [2, 3] Cruz uses a non probabilistic approach to characterize each session entering the network the burstiness constraint, which is in principle very similar to a token bucket filter. Under this assumption, a methodology is proposed to calculate per session upper bounds on delay and ....

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R.L. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," In IEEE Transactions on Information Theory, Vol. 37, No. 1, pp. 132-141, January 1991.

....acknowledge interactions with Jim Roberts at the early stages of this research. APPENDIX The purpose of this appendix is to provide a proof for Theorem 2. But first we need to establish two lemmas. Lemma 2: A necessary condition for ### ####### to be feasible is # . Proof: From [27], 28] 7] the maximum delay at smoother # (20) Suppose # # # ### # for some # . Because# # (21) And because, by assumption, # where the last equality follows from (4) Lemma 3: There exists a stochastic vector arrival process in that produces ....

R. Cruz, "A calculus for network delay, part II: Network analysis," IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 132--141, Jan. 1991.

.... to observe that, as proved in [15] universal stability implies rate stability and stability of the queueing network (the opposite is, of course, not true) Finally, we notice that an important stability result for underloaded GPS networks was obtained in [2] by applying Network Calculus [19] [20] concepts. In [2] any underloaded network of queues implementing GPS schedulers was proved to be stable when fed with leaky bucket constrained traffic flows, whenever the GPS rates assignment to flows satisfies the Consistent Relative Session Treatment (CRST) constraint. The CRST constraint ....

....if it is stable for all networks G. It is worth noting that the queueing network stability criterion provided by AQT is very tight, because it derives from a worst case analysis under a wide class of deterministic arrival patterns. In this sense, AQT is somewhat similar to Network Calculus [19] [20]. C. Fluid Models The evolution of a queueing network is traditionally described by vector equation (2) where D n is a function of both X n and the scheduling policy. An alternative expression of the evolution of the queue lengths vector can be provided in terms of the cumulative processes. ....

R.L.Cruz, "A Calculus for Network Delay, Part II: Network Analysis", IEEE Transactions on Information Theory, Vol. 37, n. 1, January 1991, pp. 132-141.

....p) #(n) q where p and q are positive integers. Of course the delay cannot decrease with more than the di#erence between the maximum and the minimum delay: max 1 #1 , # #) ##(n 1) #(n) 1(5) For an excellent and comprehensive treatise of the subject, see the work of Cruz [15, 16]. As we mentioned earlier, in the general case there is a rate mismatch between the fixed controller rate and the time variant RM cell rate. There are two cases to be considered: a) the controller rate is greater than the RM cell rate at the switch (b) the controller rate is smaller than the ....

....is di#erent. For example, the HFS delay cannot increase by more than one per time step (due to the holding action) while the VBR delay cannot decrease by more than one per time step (in order to assure that packet order is maintained) For detailed derivations of these bounds see [14, 17] or [15, 16]. The model shown in Figure 4 is a macroscopic model for the delays and their e#ects on the data rates. The delays can be viewed as the compounded delays generated in individual queues from the source to the switch, but other delay e#ects can also be modeled by this method. 2.3 Total System ....

R. L. Cruz, "A calculus for network delay, part II: Network analysis," IEEE Transactions on Information Theory, vol. 37, pp. 132--141, Jan. 1991.

....that, at any given time, a connection may be waiting in more that one shaping FIFO queue. The rationale behind this idea can be understood by considering again the piecewise linear shaping envelope for a connection, which corresponds to an ideal system of n leaky buckets connected in series [18], with parameters ## 1 ;# 1 #; ## 2 ;# 2 #; ##n;# n# (it has been shown that the order of interconnection of the leaky buckets is immaterial, therefore without loss of generality we assume # 1 ## 2 #: ## n ) Before proceeding with this discussion, we recall that in such system, the delay ....

R. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," IEEE Trans. Information Theory, pp. 121--141, Jan. 1991.

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R. L. Cruz, "A calculus for network delay, part II: Network analysis," IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 132--141, January 1991.

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R. L. Cruz, "A calculus for network delay, Part II: Network analysis," IEEE Transactions on Information Theory, vol. 37, pp. 132 -- 141, 1991.

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R. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," IEEE Transactions on Information Theory, pp. 132--141, 1991.

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R. L. Cruz, "A calculus for Network Delay, Part II: Network Analysis", IEEE Transactions on Information Theory, 37 No. 1, pp. 132-141, 1991.

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R. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," IEEE Transactions on Information Theory, pp. 132--141, 1991.

No context found.

R. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," IEEE Transactions on Information Theory, pp. 132--141, 1991.

No context found.

R. L. Cruz, "A Calculus for Network Delay, Part II: Network Analysis", IEEE Transactions on Information Theory, vol. 37, pp. 132-141, 1991.

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R. L. Cruz, "A Calculus for Network Delay, Part II: Network Analysis", IEEE Transactions on Information Theory, vol. 37, Jan 1991.

No context found.

R.L. Cruz, "A calculus for network delay, Part II: Network analysis," IEEE Trans. Inform. Theory, Vol. 37, pp. 132-141, 1991.

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R.L. Cruz, "A calculus for network delay, part II: Network Analysis" IEEE Transactions on Information Theory, Vol.37., No. 1, January 1991

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R.L. Cruz, "A calculus for network delay, Part II: Network analysis," IEEE Tran. Inform. Theory, Vol. 37, pp. 132-141, 1991.

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R.L.Cruz, "A Calculus for Network Delay, Part II: Network Analysis", IEEE Transactions on Information Theory, Vol. 37, n. 1, January 1991, pp. 132-141.

No context found.

R.L. Cruz, "A calculus for network delay, Part II: Network analysis," IEEE Tran. Inform. Theory, Vol. 37, pp. 132-141, 1991.

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R.L. Cruz, "A calculus for network delay, Part II: network analysis, " IEEE Trans. Inform. Theory, vol 37-1, pp. 132--141, January 1991.

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R. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," IEEE Transactions on Information Theory, pp. 132--141, 1991.

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