M. Andrews, A. Fernandez, M. Harchol-Balter, T. Leighton, and L. Zhang, "General dynamic routing with per-packet delay guarantees of O(distance + 1/session rate)," SIAM Journal on Computing, vol. 30, no. 5, pp. 1594--1623, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....schedule the packets that contend for the same edge simultaneously. It is known that this can be done in an optimal way if we have a static situation, i.e. no old request is cancelled and no new request arrives, and a polynomial number of time steps are allowed to compute the respective schedule [1]. However, no on line schedule with (asymptotically) optimal end to end delay is known for this situation, and no schedule with (asymptotically) optimal end to end delay is known at all for the case that we are in a dynamic scenario, that is, requests can be cancelled and new requests arrive. The ....

....of p. r; d; n) can be seen as a deadline given to the last packet to reach its destination. Afterwards, its rate can be used by other, newly injected sessions. Obviously, a protocol that guarantees an end to end delay of O(1=r i d i ) for each session i packet is worst case optimal (see also [1]) However, similar to the previous section, it is not an absolute lower bound for every individual packet. 1.2 Previous results 1.2.1 Packet switching In a pioneering paper, Leighton, Maggs and Rao [12] showed that there is an off line schedule for any simple (i.e. loop free) path collection ....

[Article contains additional citation context not shown here]

M. Andrews, A. Fernandez, M. Harcol-Balter, T. Leighton, L. Zhang. General dynamic routing with per-packet delay guarantees of O(distance + 1=session rate). In Proc. of the 38th Ann. IEEE Symp. on Foundations of Computer Science, pp. 294-302, 1997.

....that fulfill these conditions. In particular, they present a dynamic routing algorithm for the butterfly that is stable for a small constant injection rate, and they show that the expected routing time for each packet is O(log n) with n denoting the number of nodes on a level. Andrews et al. [2] investigate another, more restrictive dynamic routing model. In contrast to the stochastic and the adversarial model, the packets are injected regularly in sessions . For each session i, packets are injected at a rate r i to follow a fixed path of length d i . They describe a schedule that ....

M. Andrews, A. Fernandez, M. Harchol-Balter, T. Leighton, and L. Zhang. General dynamic routing with per-packet delay guarantees of O(distance + 1 / session rate). In Proc. of the 38th IEEE Symp. on Foundations of Computer Science (FOCS), pages 294--302, 1997.

....path, m the number of edges, and k is as suitable constant. Note that this result implies that the delay of the packets is also bounded by (D Delta log m) k , w.h.p. However, as the bound on the buffer size does not hold deterministically, any buffer of fixed size will overflow eventually. In [2], Andrews et al. introduce another dynamic routing model. In contrast to the stochastic and the adversarial model, the packets are injected regularly in sessions . For each session i, packets are injected at a rate r i to follow a fixed path of length d i . They describe a schedule that works for ....

....its destination in at most L Delta w Delta L Gamma 1 time steps. Previous results on routing with bounded buffers in leveled networks obtain stability only for constant injection rates 1 [4, 14] stochastic model) or require buffers whose size is exponential in the depth of the network [2] (adversarial model) ffl At second, we present a dynamic routing protocol for arbitrary networks with bounded buffers that is stable for any injection rate 1. The protocol requires only buffers of size D=ffl to guarantee an expected routing time of (D 2 w) ffl , and (D 2 D Delta ....

[Article contains additional citation context not shown here]

M. Andrews, A. Fern'andez, M. Harchol-Balter, T. Leighton, and Z. Zhang. General dynamic routing with per-packet delay guarantees of O(distance + 1 / session rate). In Proc. of the 38th IEEE Symp. on Foundations of Computer Science (FOCS), pages 294--302, 1997. 40

....schedule then the closest deadline first greedy strategy succeeds in routing all the messages. A number of papers consider the session model where packets are introduced at a fixed rate for each of a number of sessions . Each session consists of a given path from a source to a destination [17, 18, 2]. The aim is to schedule the packets across different links of the network with guaranteed (small) delays that depend on the rates of the sessions and the congestion on the links along the paths of the sessions. In a recent paper [3] delay requirements are added to the sessions, so that packets ....

M. Andrews, A. Fernandez, M. Harchol-Balter, F.T. Leighton, L. Zhang (1997): General dynamic routing with per-packet delay guarantees of O(distance + 1/session rate). 38th IEEE Symp. on Foundations of Computer Science.

....path, m the number of edges, and k is as suitable constant. Note that this result implies that the delay of the packets is also bounded by (D Delta logm) k , w.h.p. However, as the bound on the buffer size does not hold deterministically, any buffer of fixed size will overflow eventually. In [2], Andrews et al. introduce another dynamic routing model. In contrast to the stochastic and the adversarial model, the packets are injected regularly in sessions . For each session i, packets are injected at a rate r i to follow a fixed path of length d i . They describe a schedule that works for ....

....its destination in at most L l Delta w Delta L Gamma 1 time steps. Previous results on routing with bounded buffers in leveled networks obtain stability only for constant injection rates l 1 [4, 14] stochastic model) or require buffers whose size is exponential in the depth of the network [2] (adversarial model) At second, we present a dynamic routing protocol for arbitrary networks with bounded buffers that is stable for any injection rate l 1. The protocol requires only buffers of size D=e to guarantee an expected routing time of (D 2 w) e , and (D 2 D Delta logN ....

[Article contains additional citation context not shown here]

M. Andrews, A. Fernandez, M. Harchol-Balter, T. Leighton, and Z. Zhang. General dynamic routing with per-packet delay guarantees of O(distance + 1 / session rate). In Proc. of the 38th IEEE Symp. on Foundations of Computer Science (FOCS), pages 294--302, 1997.

....schedule the packets that contend for the same edge simultaneously. It is known that this can be done in an optimal way if we have a static situation, i.e. no old request is cancelled and no new request arrives, and a polynomial number of time steps are allowed to compute the respective schedule [1]. However, no on line schedule with (asymptotically) optimal end to end delay is known for this situation, and no schedule with (asymptotically) optimal end to end delay is known at all for the case that we are in a dynamic scenario, that is, requests can be cancelled and new requests arrive. The ....

.... every session i, packets are injected at a rate r i in order to follow a predetermined path of length d i (i.e. every 1=r i steps a session i packet is injected) Obviously, a protocol that guarantees an end toend delay of O(1=r i d i ) for each session i packet is worst case optimal (see also [1]) However, similar to the previous section, it is not an absolute lower bound for every individual packet. 1.2 Previous results 1.2.1 Packet switching In a pioneering paper, Leighton, Maggs and Rao [12] showed that there is an off line schedule for any simple (i.e. loop free) path collection ....

[Article contains additional citation context not shown here]

M. Andrews, A. Fern'andez, M. Harcol-Balter, T. Leighton, L. Zhang. General dynamic routing with per-packet delay guarantees of O(distance + 1=session rate). In Proc. of the 38th Ann. IEEE Symp. on Foundations of Computer Science, pp. 294-302, 1997.

....for streams as for batch delivery, with the congestion C replaced by maxf1=R i g in the periodic model, and by the maximum overload period in the adversarial models. See also Broder, Frieze, and Upfal [6] for related work. Very recently, Andrews, Fern andez, HarcholBalter, Leighton, and Zhang [2] announced an O(D i 1=R i ) centralized algorithm for the periodic traffic model. Our results. The main result of this paper is to show a randomized local control algorithm that for any ffl 0 with probability 1 Gamma 1=poly(N) delivers packets in time O Gamma C D (log 1 ffl N) ....

M. Andrews, A. Fern'andez, M. Harchol-Balter, F.T. Leighton, and L. Zhang. General dynamic routing with per-packet delay guarantees of O( distance + 1 = session rate ). Unpublished manuscript, February 1997.

....once every 1=r i time steps and travel along specified length d i paths. A distributed routing algorithm in [35, 36] guarantees a delay of 2d i =r i for session i packets, provided that edges are used with rate 1; this bound is improved to O(1=r i d i ) via a centralized algorithm, in [3]. By creating a different session for each packet, these results can be used to route each packet p i to its destination in time O(c i d i ) where c i is the maximum congestion on the path of p i . Note that while these results provide per packet delay bounds, they do not accommodate arbitrary ....

M. Andrews, A. Fernandez, M. Harchol-Balter, F.T. Leighton, L. Zhang (1997): General dynamic routing with per-packet delay guarantees of O(distance + 1/session rate). 38th IEEE Symp. on Foundations of Computer Science.

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M. Andrews, A. Fernandez, M. Harchol-Balter, T. Leighton, and L. Zhang, "General dynamic routing with per-packet delay guarantees of O(distance + 1/session rate)," SIAM Journal on Computing, vol. 30, no. 5, pp. 1594--1623, 2000.

....end to end delay derived for WFQ in [13] for a given session i is O(K i 1= i ) where K i is the number of links of the session path and i is the session rate. However, it has been shown that bounds of O(K i 1= i ) can be achieved with simple deadline based randomized scheduling algorithms [3, 4]. In [4] Andrews and Zhang, proposed the Coordinated Earliest Deadline First (CEDF) scheduling policy, for which they derived the analytical bound, and showed by simulation that the bound difference between WFQ and CEDF could be observed in simple setups. For the simulation they used a second ....

M. Andrews, A. Fernandez, M. Harchol-Balter, T. Leighton, and L. Zhang, "General dynamic routing with per-packet delay guarantees of O(distance + 1/session rate)," in Proc. of the 38th Symp. on Found. of Comp. Sc., (Miami Beach, FL), pp. 294--302, Oct. 1997.

....Clock [11] and Frame Based Fair Queueing [9, 10] Initially, it is not clear whether it is possible to improve these bounds, since it seems natural that a packet might need to wait O(1=r i ) steps at each switch. However, these bounds are far from the lower bound of Omega Gamma d i b i =r i ) [2]. In previous related work, Rabani et al. 8, 5] obtain algorithms with delay bounds in which the path distance and the inverse of the rate are added instead of multiplied. However, the bound obtained depend on the values D = max i (d i ) and R = max i (1=r i ) and, therefore, the delay bound for ....

....(1=r i ) and, therefore, the delay bound for one session can depend strongly on the parameters for other sessions. Furthermore, the algorithms they present allow some packets to be lost. The first proof that the lower bound of Omega Gamma d i b i =r i ) can actually be achieved is presented in [2]. Here it is shown that there exists a periodic schedule that routes all the packets to their destinations in O(d i b i =r i ) steps. Furthermore, the schedule guarantees that, once a packet starts moving, it never waits more than a constant number of steps in each intermediate queue. This ....

[Article contains additional citation context not shown here]

Matthew Andrews, Antonio Fern'andez, Mor Harchol-Balter, Tom Leighton, and Lisa Zhang, "General dynamic routing with per-packet delay guarantees of O(distance + 1/session rate)," Unpublished manuscript.

No context found.

M. Andrews, A. Fern'andez, M. Harchol-Balter, and T. Leighton, L. Zhang, "General Dynamic Routing with Per-Packet Delay Guarantees of O(distance + 1/session rate)," Proc. of 38th FOCS, pp. 294--302, 1997.

No context found.

M. Andrews, A. Fern'andez, M. Harchol-Balter, F.T. Leighton, and L. Zhang. General dynamic routing with per-packet delay guarantees of O( distance + 1 = session rate ). Unpublished manuscript, February 1997.

No context found.

M. Andrews, A. Fern'andez, M. Harchol-Balter, and T. Leighton, L. Zhang, "General Dynamic Routing with Per-Packet Delay Guarantees of O(distance + 1/session rate)," In Proc. of 38th FOCS, pp. 294--302, 1997.

No context found.

M. Andrews, A. Fern'andez, M. Harchol-Balter, and T. Leighton, L. Zhang, "General Dynamic Routing with Per-Packet Delay Guarantees of O(distance + 1/session rate)," Proc. of 38th FOCS, pp. 294--302, 1997.

No context found.

M. Andrews, A. Fernandez, M. Harchol-Balter, F.T. Leighton, and L. Zhang (1997): General dynamic routing with per-packet delay guarantees of O(distance + 1/session rate). Proc.38th IEEE Symp. on Foundations of Computer Science, pp. 294--302.

No context found.

M. Andrews, A. Fernandez, M. Harchol-Balter, T. Leighton, and L. Zhang. General dynamic routing with per-packet delay guarantees of O(distance + 1 / session rate). In Proc. of the 38th IEEE Symp. on Foundations of Computer Science (FOCS), pages 294--302, 1997.

No context found.

M. Andrews, A. Fernandez, M. Harchol-Balter, and T. Leighton, L. Zhang, \General Dynamic Routing with Per-Packet Delay Guarantees of O(distance + 1/session rate)," In Proc. of 38th FOCS, pp. 294-302, 1997.

No context found.

M. Andrews, A. Fern'andez, M. Harchol-Balter, and T. Leighton, L. Zhang, "General Dynamic Routing with Per-Packet Delay Guarantees of O(distance + 1/session rate)," Proc. of 38th FOCS, pp. 294--302, 1997.

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