Cruz R. L. and Okino C. M. Service guarantees for window flow control. In 34th Allerton Conf of Comm., Cont., and Comp. Monticello, IL, Oct 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....# r.How ever, the same cannot be said for PSRG, since it gives a delay from backlog bound; there may be cases where the only information available on the aggregate input is a bound on sustainable rate #, with # r. In such cases, there are probably other mechanisms (such as window flow control [18]) to prevent buffer overflow; here, it is useful to be able to bound e # as in Theorem V.2. VII. CONCLUSION We have introduced a new technique, based on minmax algebra, for the analysis of non FIFO PSRG nodes. We have shown that both delay from backlog bounds and delay bound from arrival curve ....

Cruz R. L. and Okino C. M, "Service guarantees for window flow control," in 34th Allerton Conf of Comm., Cont., and Comp. Monticello, IL, Oct 1996.

....node e is positive, the number of cells of flow r that are output by the node is fi(t) The backlog for flow r at node e at some time instant is defined as the number of cells of flow r, which have entered node e and did not depart yet. The strict service curve property was defined for example in [5] and is an abstraction of the generalized processor sharing concepts introduced in [5] The following theorem is a new variant of classical results in [3] 1] 2] and [6] Theorem 2.1: Consider a node that receives an input connection, with a buffer large enough to avoid discarding data. ....

....The backlog for flow r at node e at some time instant is defined as the number of cells of flow r, which have entered node e and did not depart yet. The strict service curve property was defined for example in [5] and is an abstraction of the generalized processor sharing concepts introduced in [5]. The following theorem is a new variant of classical results in [3] 1] 2] and [6] Theorem 2.1: Consider a node that receives an input connection, with a buffer large enough to avoid discarding data. Assume that the node offers a strict service curve fi and that the input connection has an ....

R. L. Cruz and C. M. Okino, "Service guarantees for window flow control," in Proc. 34th Allerton Conf Comm., Cont., and Comp. Monticello, IL, Oct. 1996.

....Based on this, we provide a couple of rules for service curve allocation among a concatenation of servers. We note that the importance of the role of the (min; algebra in deterministic traffic 2 regulation and service guarantees is also recognized by Agrawal and Rajan [2] Cruz and Okino [15] and Le Boudec [4] In particular, a division operator in the (min; algebra is defined in [2, 15] Such an operator provides a simple representation of performance bounds and output burstiness. Extensions of such an operator can be found in Chang [9] and Cruz [13] For the window flow control ....

....of servers. We note that the importance of the role of the (min; algebra in deterministic traffic 2 regulation and service guarantees is also recognized by Agrawal and Rajan [2] Cruz and Okino [15] and Le Boudec [4] In particular, a division operator in the (min; algebra is defined in [2, 15]. Such an operator provides a simple representation of performance bounds and output burstiness. Extensions of such an operator can be found in Chang [9] and Cruz [13] For the window flow control problem, Cruz and Okino [15] considered a more detailed model than the one in Agrawal and Rajan [2] ....

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R.L. Cruz and C.M. Okino, "Service guarantees for window flow control," Proc. 34th Allerton Conf. on Comm., Cont. & Comp., Monticello, IL, Oct. 1996.

....the optimal VBR trunk characteristics. 0018 9448 98 10.00 1998 IEEE 1088 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998 II. BACKGROUND:NETWORK CALCULUS In this section we recall a few definitions and results, which we collectively call network calculus [8] 10] 11] [18], 19] We consider wide sense increasing functions of time, with nonnegative, possibly infinite values (we say that function is wide sense increasing, also called nondecreasing, when for all ) For two wide sense increasing functions and , define by (1) This operation is called min plus ....

....is continuous (no batch departure) then . Network calculus results give computational rules for bounding virtual delays and backlog for arbitrary systems that represent networks. We first need the definition of service curve. We say that offers to the flow a service curve if and only if [10] [18] (3) This is equivalent in practice to saying that for all , there exists some , with , such that The definition of a service curve is an abstraction of the strict service curve concept defined by Cruz in [5] A GPS scheduler [1] 2] with rate guarantee to a flow offers this flow a service ....

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R. L. Cruz and C. M. Okino, "Service guarantees for window flow control," in 34th Allerton Conf. Communication, Control, and Computing (Monticello, IL, Oct. 1996).

....higher, and may not even be bounded. Therefore, the class of LR servers is larger than the class of servers based on bounds derived with respect to a backlogged period. Since the first presentation of this work in [24] the notion of the busy period was extended to non linear service curves in [7, 1]. Finaly, Hung and Kesidis [15] presented a method for computation of delay bounds in schedulers. Their results are restricted to isolated packets in a single scheduler and the extension to arbitrary traffic distributions and networks of servers is not straightforward. In addition, their model is ....

R.L. Cruz and C.M. Okino. Service guarantees for window flow control. In Proceedings of the 34th Allerton Conference on Communication Control and Computing, October 1996.

....of the arrival traffic. If the cost function for the VBR trunk is linear, then we find an explicit algorithm for computing the optimal VBR trunk characteristics. II Background: Network Calculus In this section we recall a few definitions and results, which we collectively call network calculus [18, 8, 11, 10, 19]. We consider wide sense increasing functions of time, with non negative, possibly infinite values (we say that function fl( is wide sense increasing , also called non decreasing , when fl(s) fl(t) for all s t) For two wide sense increasing functions fl 1 and fl 2 , define fl 1 Omega fl 2 ....

....batch departure) then R (t d(t) R(t) Network calculus results give computational rules for bounding virtual delays and backlog for arbitrary systems that represent networks. We first need the definition of service curve. We say that S offers to the flow a service curve fi if and only if [18, 10] R R Omega fi (3) This is equivalent in practice to saying that for all t 0, there exists some t 0 0, with t 0 t, such that R (t) Gamma R(t 0 ) fi(t Gamma t 0 ) The definition of a service curve is an abstraction of the strict service curve concept defined by Cruz in [5] A GPS ....

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Cruz R. L. and Okino C. M. Service guarantees for window flow control. In 34th Allerton Conf of Comm., Cont., and Comp. Monticello, IL, Oct 1996.

....in conjunction with rate based flow control at access points and appropriate scheduling algorithms within the network, windowbased flow control can provide performance guarantees and may offer an attractive alternative to open loop rate based flow control within the network. In an earlier paper [9], we studied the flow of traffic on a single stream subject to window flow control, where cross traffic was modelled through constraints on servers. In this paper, we explicitly model multiple streams in a general mesh network, where each stream is subject to window flow control and packet ....

....i;f (v) S h i (t Gamma v) R h Gamma1 i;f S h i (t) 23) Thus, 22) and (23) contradict (21) and so R h i (t) R h Gamma1 i;f S h i (t) if T h i [t] K h i . 24) Finally, 20) and (24) prove (17) for the hypothesis H(t 1) Pi 5 Closing Remarks Given the results of [9][1] 4] 5] one might expect that a window size of K h i;old is sufficient to guarantee that stream (i; h) is guaranteed the service curve S h i , where K h i;old = max x f S h i (x) Gamma S h i S h i S h i;loop (x)g : 25) Comparing this to Theorem 1, it can be shown that K ....

R. L. Cruz and C. M. Okino, "Service Guarantees for Window Flow Control," Proc. 34 th Allerton Conf. on Comm., Cont. & Comp., Monticello, IL, Oct. 1996.

....in using a simple calculus based on the convolution operation. It is apparent that the framework is related to the theory of linear filtering, where the concept of a service curve here is somewhat analogous to the impulse response of a linear system. This analogy is discussed at more length in [10]. We have analyzed unicast guaranteed sessions as a series of service curve elements, and obtained bounds on end to end delay and buffering requirements. This paper also suggested how regulation may be used with the network to reduce resource requirements for a session without compromising ....

R. L. Cruz and C. M. Okino. Service guarantees for window flow control. Proceedings of the 34th Allerton Conference on Communication, Control, & Computing , Monticello, IL, October, 1996.

....algorithm [31] Our proposal differs in that delay jitter is not necessarily forced to zero inside a virtual path, to allow for efficient statistical multiplexing. The remainder of this paper is organized as follows. In the next section, we refine the so called network calculus [1] 5] 6] 7] 10][11][20] 26] to address delay jitter, introducing the concept of a maximum service curve. This will enable us to obtain a very simple and unified result for the output burstiness of a general network element. We also include some results which address the issue of aggregation. All of these results are ....

....the traffic stream. Then for all n there holds R in (n Gamma d) R out (n) R in (n Gamma d) where d = minf Delta : Delta 0 and S(x) b ffi Delta (x) for all x 0g : and d = maxfx : x 0 and S(x) 0g : We note that the lower bound in Theorem 1 was reported before [1] 5][11][20] 10] 26] although the proof here is even more streamlined, due to the representation here of d, which can be interpreted graphically as the largest horizontal distance between b and S. Proof: We have R out (n) R in S(n) R in (b ffi d ) n) R in b) ffi d (n) R in ....

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R. L. Cruz and C. Okino. Service guarantees for window flow control. Proceedings of the 1996 Allerton Conference on Communication and Control, Monticello, IL, October, 1996.

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Cruz R. L. and Okino C. M. Service guarantees for window flow control. In 34th Allerton Conf of Comm., Cont., and Comp. Monticello, IL, Oct 1996.

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Cruz R. L. and Okino C. M. Service guarantees for window flow control. In 34th Allerton Conf of Comm., Cont., and Comp. Monticello, IL, Oct 1996.

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R.L. Cruz and C.M. Okino, "Service guarantees for window flow control," Proc. 34th Allerton Conf. on Comm., Cont. & Comp., Monticello, IL, Oct. 1996.

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