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.

....input traffic specification, i.e. packet arrival behavior at its source node. Specifying modeling a real time application s traffic pattern is a challenging problem especially when source traffic has variable bit rate (VBR) characteristics, e.g. MPEG coded video. The leaky bucket model [1, 2] and the (X min ; X ave ; I ; Smax ) model [3, 4] are prototypical example input traffic specifications. In the leaky bucket model, the amount of connection i s traffic generated during time interval [t; is assumed to be upper bounded by oe i ae i ( Gamma t) That is, the size of an ....

R. L. Cruz, "A calculus for network delay, part II: network analysis," IEEE Trans. on Information Theory, vol. 37, no. 1, pp. 132--142, January 1991.

....al. 13] also derived a statistical bound on the end to end delay by applying the Exponentially Bounded Burstiness (E.B.B. process model [14] to Generalized Processor Sharing (GPS) networks in a similar way as Parekh and Gallager [15, 16] derived a deterministic bound using the leaky bucket model [17, 18]. Although their work is theoretically attractive, it is not clear whether it can be applied to real life systems since their scheme assumes an infinite buffer at each node. Moreover, the implementation complexity of PGPS must be resolved before it can be used for high speed networks like ATM ....

R. L. Cruz, "A calculus for network delay, part II: network analysis," IEEE Trans. on Information Theory, vol. 37, no. 1, pp. 132--142, January 1991.

....TCRM (1) has efficiency close to PGPS in terms of channel admissibility; 2) is simple enough to operate in a highspeed switching environment like ATM networks; 3) requires only a very small buffer space for each realtime channel. 2 The Proposed Scheme We assume that the leaky bucket model [14, 15] is given as the input traffic description. The leaky bucket model, simply denoted by (oe i ,ae i ) is to place a smoothing buffer (leaky bucket regulator) of size oe i and token generation rate ae i at the network entrance so that the burstiness of input traffic into the network may be reduced, ....

R. L. Cruz, "A calculus for network delay, part II: network analysis", IEEE Trans. on Information Theory, vol. 37, pp. 132--142, Jan. 1991.

....knowledge, this problem has not been treated previously. The remainder of the paper is organized as follows. In Section 2, we review the related work in the fields of m2p architectures and end to end bounds in FIFO networks. In Section 3, we recall the main results of the Network Calculus [6] [7], 8] that we use to obtain a bound on the end to end delay. In Section 4, we emphasize the difficulty to directly derive an end to end delay bound for a FIFO network from the concept of service curve introduced in the Network Calculus. In Section 5, we study the maximum end to end delay in the ....

....applies to FIFO networks with a general architecture, but it is restricted to the case of constant bit rate sources. In the present work, we concentrate on a specific architecture, the m2p architecture, however with variable bit rate sources. III. NETWORK CALCULUS The Network Calculus [13] 6] [7], 8] 14] is an analytical method to derive deterministic bounds on end to end delays and backlogs. The Network Calculus has been developed both for continuous time [13] and discrete time [8] We use here the continuous version that is better suited for a fluid flow analysis. We present, in the ....

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

....node only. Sources are assumed to be leaky bucket constrained. In [13] we provide an accurate upper bound on the end to end delay for m t p networks where the scheduling policy is FIFO. In [12] a corresponding admission procedure is derived. These results are obtained using the Network Calculus [7,8,2,4]. We first briefly recall the bases of Network Calculus and discuss the end to end delay bound and the corresponding admission control procedure. 2.1. Network Calculus Network Calculus provides deterministic bounds on end to end delays and backlogs. A source is modeled through an arrival curve ....

R. Cruz, "A calculus for Network Delay, Part II: Network Analysis", IEEE Transactions On Information Theory, 37(1):132--141, January 1991.

....is grouped in connections or sessions. Each session has a source and a destination node, which are the corresponding source and destination nodes of each packet belonging to the session. The flow of new packets arriving to a session i is described by a pair ( i ; i ) as introduced by Cruz [6, 7], where 0 i 1 is the rate and i 1 is the burst size of the session. If A i (t 1 ; t 2 ) denotes the number of packets arrived to the network belonging to session i in the time interval (t i ; t 2 ] then A i (t 1 ; t 2 ) i i (t 2 t 1 ) From this, it can be simply observed that ....

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

....show that this results still holds (for leaky buckets, not in general) in presence of packetization effects. The theory in [2] 3] is for discrete time systems with constant packet sizes, and thus applies without restriction to ATM systems. In contrast, and unlike the original results in [1] [12], the theory of greedy shapers in [4] 5] applies to continuous time and flows with variable packet size, but does not account for packetization effects. In this context, the greedy shaper characterized by Equation (2) outputs a continuous stream of bits, not entire packets; see for example ....

.... for the example in Figure 7, the tandem of packetized greedy shapers with curves # and # does not have an # smooth output, therefore it cannot be equivalent to the packetized greedy shaper with curve min(#, #) Thus, we have proven that the conservation and decomposition properties established in [12] for (b, r) regulators holds for the more usually accepted definition of buffered leaky bucket controller. However, we have also found that, unlike the results for constant size packets in [3] 4] 5] we cannot, in general, extend this property to any arbitrary arrival curve. 9 V. CONCLUSION ....

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

....a review of packet scheduling. Many of these results can be cast into a common framework coined network calculus , which we explain in this section. In short, network calculus can be viewed as the application of min and max algebra to flow problems. It was pioneered by Chang [14] and Cruz [15] [16], and found its final form in subsequent work by the same authors and by Agrawal, Le Boudec and Rajan [17] 18] 19] A comprehensive treatment can be found in two textbooks [5] 6] We first introduce network calculus on an example, then we review applications to integrated services, ....

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

....in [3, 4, 5] In this paper, we focus on the property, already mentioned, that greedy shapers keep arrival constraints. The theory in [2, 3] is for discrete time systems with constant packet sizes, and thus applies without restriction to ATM systems. In contrast, and unlike the original results in [1, 9], the theory of greedy shapers in [4, 5] applies to continuous time and flows with variable packet size, but does not account for packetization effects. In this context, the greedy shaper characterized by Equation (2) outputs a continuous stream of bits, not entire packets; see for example Figure ....

....to a packetized greedy shaper with cumulative packet length # and shaping curve #. Assume that # and # are concave with ### # # # ### and ### # # # ### . Then the output flow is still constrained by the original arrival curve #. Thus, we have proven that the conservation property found in [9] for ### ## regulators holds for the more usually accepted definition of buffered leaky bucket controller. However, we have also found that, unlike the results for constant size packets in [3, 4, 5] we cannot, in general, extend this property to any arbitrary arrival curve. 6 Conclusion We have ....

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

....p (s) d p (t Gamma s) fl p for all t s 0: 3) It should be noted that the class of bounded burstiness inputs, is one of the few classes for which bounds on important design parameters, such as buffer levels, can be determined. The use of bounded burstiness inputs has been introduced in Cruz [2, 3]. The results presented here will be derived for systems with linear deterministic and bounded burstiness inputs. Let ae p;i : d p p;i be the load factor imposed by part type p on the machine at the i th stage of its route. An important role is played by the load factor ae m at machine m, ....

....course some time is lost due to repeated switchings. Machine i regulator buffer regulated buffer Figure 3: A regulator buffer constrains the input of parts available to machine i . We will consider a third approach, which implements system elements called regulators; see also Cruz [3]. Such regulators can be implemented as in Figure 3, by splitting the buffer into two (virtual) components, a regulator buffer and a regulated buffer. Only parts in the regulated buffer are available for processing by the machine. The flow of parts into the regulated buffer from the regulator ....

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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, January 1991.

....Services architecture (or IntServ) and the Differentiated Services architecture (or DiffServ) 1. 3 Integrated Services architecture The Integrated Services (IntServ) architecture was developed within the IETF in the mid nineties [15, 118, 115] and has its roots in earlier research results [5, 79, 27, 28, 38]. The basic premise in IntServ is that applications have hard network performance requirements, and they cannot operate effectively unless these requirements are met. For instance, an IP telephony application may require a maximum end to end delay of 200msec for each packet, while a video ....

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.

....probability that the QoS parameters are violated. In a deterministic service, all the packets are guaranteed to be delivered within the specified QoS bounds [3, 8, 9, 10, 11, 12, 13, 14] In such services, the trac characteristics are parameterized by a deterministic trac model. In the ( p) model [20, 21], for an interval of length , the source is constrained to transmitting no more than p bits. The ( p model [3] which is an extension of the ( p) model, defines a two dimensional array of (k,Pk) pairs resulting in a piecewise concave function. The discrete model [22] uses four parameters ....

R. L. Cruz, "A calculus for network delay, part II: Network analysis," IEEE Trans- actions on Information Theory, vol. 37, no. 1, pp. 121 141, Jan. 1991.

....provides general stochastic bounds at each node of a network. A central problem shared by all network calculi is of determining the conditions under which a network is stable, meaning that the queue length at each element of the network is bounded according to some appropriate metric [5] [11]. It turns out that network stability is easy to establish only for feed forward routing networks, i.e. networks where routes do not create cycles of interdependent packet flows. Such network are stable if and only if the traffic load (utilization) at each element is smaller than one [5] 11] ....

....[11] It turns out that network stability is easy to establish only for feed forward routing networks, i.e. networks where routes do not create cycles of interdependent packet flows. Such network are stable if and only if the traffic load (utilization) at each element is smaller than one [5] [11]. This condition is known as the throughput condition [28] The case of non feed forward networks is generally much more complicated, with only a few notable exceptions (e.g. 19] While the throughput condition remains necessary for the stability of such networks, it is no longer sufficient. A ....

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

....provides general stochastic bounds at each node of a network. A central problem shared by all network calculi is of determining the conditions under which a network is stable, meaning that the queue length at each element of the network is bounded according to some appropriate metric [5] [10]. It turns out that D. Starobinski and M. Karpovsky are with the ECE department at Boston University. E mail: fstaro,markkarg bu.edu. L. Zakrevski is with the ECE department at the New Jersey Institute of Technology. E mail: zakr adm.njit.edu. The work of the second and third authors was ....

....under Grant MIP 9630096 network stability is easy to establish only for feed forward routing networks, i.e. networks where routes do not create cycles of interdependent packet flows. Such network are stable if and only if the traffic load (utilization) at each element is smaller than one [5] [10]. This condition is known as the throughput condition [26] The case of non feed forward networks is generally much more complicated, with only a few notable exceptions (e.g. 17] While the throughput condition remains necessary for the stability of such networks, it is no longer sufficient. A ....

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

....provides general stochastic bounds at each node of a network. A central problem shared by all network calculi is of determining the conditions under which a network is stable, meaning that the queue length at each element of the network is bounded according to some appropriate metric [5] [10]. It turns out that network stability is easy to establish only for feed forward routing networks, i.e. networks where routes do not create cycles D. Starobinski and M. Karpovsky are with the ECE department at Boston University. E mail: staro,markkar bu.edu. L. Zakrevski is with the ECE ....

....E mail: zakr adm.njit.edu. The work of the second and third authors was supported in part by the National Science Foundation under Grant MIP 9630096 of interdependent packet flows. Such network are stable if and only if the traffic load (utilization) at each element is smaller than one [5] [10]. This condition is known as the throughput condition [27] The case of non feed forward networks is generally much more complicated, with only a few notable exceptions (e.g. 18] While the throughput condition remains necessary for the stability of such networks, it is no longer sufficient. A ....

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

....as the worst case fairness index (denoted by j WF ) for that scheme. 4. Efficient latency tuning characteristics: One of the proposed frameworks for lossless transport of real time data with guaranteed delay is to regulate the session s traffic at the network edge by a leaky bucket regulator [10, 11, 12], and to guarantee a service curve, defined by latency and rate [6] at each of the intermediate nodes. It is then possible to guarantee an upper bound on the end to end delay. In analogy with the leaky bucket regulator which enforces an upper affine envelope on the allowable volume of traffic of ....

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

....a review of packet scheduling. Many of these results can be cast into a common framework coined network calculus , which we explain in this section. In short, network calculus can be viewed as the application of ### and ### algebra to flow problems. It was pioneered by Chang [14] and Cruz [15] [16], and found its final form in subsequent work by the same authors and by Agrawal, Le Boudec and Rajan [17] 18] 19] A comprehensive treatment can be found in two textbooks [5] 6] We first introduce network calculs on an example, then we review applications to integrated services, ....

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

....shapers, traffic between adjacent switching nodes can not be isolated, thus within the network there could exist very complicated global traffic feedback which results from loops consisting of route segments of different communication sessions. Such feedback phenomenon has been pointed out by Cruz [3] and Parekh [15] for communication networks and by Kumar [11] for manufacturing systems. Consequently, the local EDF schedulability condition in Eq. 2) or (4) can not be directly extended into a global (end to end) schedulability condition for RC Method 2, even the deadlines are assigned in the ....

R. Cruz, "A calculus for network delay, part II: Network analysis," IEEE Tran. Information Theory, vol. 37, pp. 121-141, 1991.

....20, Finally, admission control has been investigated as well [12, 14, 18] Some of the proposals have been implemented and deployed, for example, NetEx, RSVP, and Tenet. NetEx [2, 3] is a good working example of implemented system based on extensions of Cruz s methodology for delay analysis [4, 5, 6]. RSVP [21] has been proposed as the signaling and resource reservation in the Integrated Services architecture. Tenet Scheme II [22] uses a two pass resource allocation scheme, with extensive functionality needed for multi party communication. However, one common point of all these systems is ....

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

....et al. 103] also derived a statistical bound on the end to end delay by applying the ExponentiallyBounded Burstiness (E.B.B. process model [94] to Generalized Processor Sharing (GPS) networks in a similar way Parekh and Gallager [65, 66] derived a deterministic bound using the leaky bucket model [14, 15]. Although their work is theoretically attractive, it is not clear whether it can be applied to real life systems since their scheme assumes an infinite buffer at each node. In addition, the implementation complexity of PGPS must be resolved for use in high speed networks like ATM [51] Effective ....

....service in an ISPN. We first adopt the concept of traffic envelope to describe the source traffic characteristics, which is necessary for admission control of new connection requests and assessment of their resource reservation requirements. Instead of using a deterministic traffic envelope [14, 15, 24, 44], we employ a statistical traffic envelope derived from exploiting the statistical characteristics of source traffic. We take a direct approach based on the Central Limit Theorem rather than a bounding approach taken in previous work. The statistical traffic envelope enables us to greatly reduce ....

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R. L. Cruz, "A calculus for network delay, part II: network analysis," IEEE Trans. on Information Theory, vol. 37, no. 1, pp. 132--142, January 1991.

....during [t 1 ; t2 ) whose assigned paths contain edge e, is at most r(t2 Gammat 1) w. Formally, for each e 2 E and t = 0; 1; 2; we restrict the adversary by X P :e2P AP (t1 ; t2 ) r(t2 Gamma t1 ) w: 4) Assumption (4) is a generalization of the assumption considered by Cruz in [7], where for each path P the associated arrival process was assumed to be sublinear: for some rP ; bP and for all t 0 AP (t) rP t bP : 5) We will say that the network has path wise constant arrival rates if (5) holds and if, in addition, for some b 0 P 0 and for all t 0 AP (t) rP t ....

R. Cruz, "A calculus for network delay, part II: network analysis," IEEE Trans. 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, no. 1, pp. 132--141, January 1991.

....(with A(0) 0) conforms to a function f , called an envelope, if A(t) Gamma A(s) f(t Gamma s) 8s t: Without loss of generality, an envelope f can be assumed to be subadditive [8] i.e. f(s) f(t Gamma s) f(t) for all s t. Using this characterization, a calculus is developed in [12] [13] to compute deterministic performance measures, such as bounds on delay and bounds on queue length. Traffic regulation addresses the problem of modifying a traffic stream so that it conforms to a subadditive envelope f . The problem of traffic regulation was treated systematically in [10] There ....

....a definition, f G is in F . One may view f G as a special case of F G for some F 2 F with F (0; t) f(t) for all t and F (s; t) 1, for all t and s 0. Thus, the results in Lemma II.3 still hold. III. CONSTRAINED TRAFFIC REGULATION Given a sequence A 2 F 0 , it is defined in [12] [13] that A conforms to the (static) upper envelope f 2 F 0 if A(t) Gamma A(s) f(t Gamma s) for all s t. It is also shown in [10] 1] that the optimal traffic regulator that generates output traffic conforming to a subadditive envelope f is a linear time invariant filter with the impulse response ....

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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, vol. 37, no. 1, 132-141, January 1991. 184

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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.

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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, 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.

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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

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.

No context found.

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.

No context found.

R.L. Cruz, "A calculus for network delay, Part II: network analysis, " IEEE Trans. Inform. Theory, vol 37-1, pp. 132--141, January 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. Cruz, "A Calculus for Network Delay, Part II: Network Analysis", IEEE Transactions on Information Theory, Vol. 37, No. 1, January

No context found.

R. Cruz, "A Calculus for Network Delay, Part II: Network Analysis," IEEE Transactions on Information Theory, pp. 132--141, 1991. 4

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