Albert G. Greenberg

Stability of binary exponential backoff

Binary exponential backoff is a randomized protocol for regulating transmissions on a multiple-access broadcast channel. Ethernet, a local-area network, is built upon this protocol. The fundamental theoretical issue is stability: Does the backlog of packets awaiting transmission remain bounded in time, provided the rates of new packet arrivals are small enough? It is assumed n ≥ 2 stations share the channel, each having an infinite buffer where packets accumulate while the station attempts to transmit the first from the buffer. Here, it is established that binary exponential backoff is stable if the sum of the arrival rates is sufficiently small. Detailed results are obtained on which rates lead to stability when n = 2 stations share the channel. In passing, several other results are derived bearing on the efficiency of the conflict resolution process. Simulation results are reported that, in particular, indicate alternative retransmission protocols can significantly improve performance.

How fair is fair queuing

Fair Queuing is a novel queuing discipline with important applications to data networks that support variable-size packets and to systems where the cost of preempting jobs from service is high. The disciphne controls a single server shared by N job arrival streams with each stream allotted a separate queue. After every job completion, the server is assigned to serve, without possibihty of interruption, the job at the head of one of the queues (as soon as at least one job appears in the system). Fair Queuing is designed to handle arbitrary job arrival sequences with essentially no a priori knowledge of their attributes. such that each stream receives its Lfam share of serwce. In this paper, we consider two variants of the fair queuing discipline, and rigorously establish their fairness wa sample path comparisons with the head-of-line processor sharing disclphne, a mathematical idealization that prowdes a fairness paradigm. An efficient Implementation of one of the fair queuing disciplines is presented, In passing, a new, fast method for simulating processor sharing is derived. Simulation results are presented to further explore the comparison between fair queuing and processor sharing.

Hardware-efficient fair queueing architectures for high-speed networks

In emerging communication networks (B-ISDN based on asynchronous transfer mode (ATM) technology), a single link may carry traffic for thousands of connections with different traffic parameters and quality-of-service requirements. High-speed links, coupled with small packet/cell sizes, require efficient switch architectures that can handle cell arrivals and departures every few microseconds, or faster. This paper presents a collection of self-clocked fair queueing (SCFQ) architectures amenable to efficient hardware implementation in network switches. Exact and approximate implementations of SCFQ efficiently handle a moderate range of connection bandwidth parameters, while hierarchical arbitration schemes scale to a large range of throughput requirements. Simulation experiments demonstrate that these architectures divide link bandwidth fairly on a small time scale, preserving connection bandwidth and burstiness properties.

Deflection routing in hypercube networks

An approximate analysis of the transient and steady state behavior of deflection routing in hypercube networks is presented, under a uniform traffic model. In deflection routing congestion causes packets admitted to the network to be temporarily misrouted rather than buffered or dropped. The approximations show that deflection routing performs remarkably well in hypercube networks, for small as well as large networks and for the whole range from light to heavy load. Simulations suggest that the approximations are quite accurate. >

A lower bound on the time needed in the worst case to resolve conflicts deterministically in multiple access channels

A problem related to the decentralized control of a multiple access channel is considered: Suppose <italic>k</italic> stations from an ensemble of <italic>n</italic> simultaneously transmit to a multiple access channel that provides the feedback 0, 1, or 2+, denoting <italic>k</italic> = 0, <italic>k</italic> = 1, or <italic>k</italic> ≥ 2, respectively. If <italic>k</italic> = 1, then the transmission succeeds. But if <italic>k</italic> ≥ 2, as a result of the conflict, none of the transmissions succeed. An algorithm to <italic>resolve</italic> a conflict determines how to schedule retransmissions so that each of the conflicting stations eventually transmits singly to the channel. In this paper, a general model of deterministic algorithms to resolve conflicts is introduced, and it is established that, for all <italic>k</italic> and <italic>n</italic> (2 ≤ <italic>k</italic> ≤ <italic>n</italic>), &OHgr;(<italic>k</italic>(log <italic>n</italic>)/(log <italic>k</italic>)) time must elapse in the worst case before all <italic>k</italic> transmissions succeed.

Sharp approximate models of deflection routing in mesh networks

A device which scans the density of successive in crements of a cigarette rod produced in a cigarette making machine, is used to control the mean weight of the tobacco in that rod with a view toward maintaining substantially constant the number of cigarettes produced having a weight falling below a preselected level. Signals produced by the scanner are used to control the quantity of excess tobacco removed from the filler stream formed therein by a trimming device, thereby controlling the quantity of tobacco placed in said rod.

Algorithms for unboundedly parallel simulations

New methods are presented for parallel simulation of discrete event systems that, when applicable, can usefully employ a number of processors much larger than the number of objects in the system being simulated, Abandoning the distributed event list approach, the simulation problem is posed using recurrence relations. We bring three algorithmic ideas to bear on parallel simulation: parallel prefix computation, parallel merging, and iterative folding. Efficient parallel simulations are given for (in turn) the G/G/l queue, a variety of queueing networks having a global first come first served structure (e.g., a series of queues with finite buffers), acyclic networks of queues, and networks of queues with feedbacks and cycles. In particular, the problem of simulating the arrival and departure times for the first N jobs to a single G/G/l queue is solved in time proportional to N/P + log P using P processors.

Estimating the multiplicities of conflicts to speed their resolution in multiple access channels

New, improved algorithms are proposed for regulating access to a multiple-access channel, a common channel shared by many geographically distributed computing stations. A conflict of <italic>multiplicity n</italic> occurs when <italic>n</italic> stations transmit simultaneously to the channel. As a result, all stations receive feedback indicating whether <italic>n</italic> is 0, 1, or ≥2. If <italic>n</italic> = 1, the transmission succeeds; whereas if <italic>n</italic> ≥ 2, all the transmissions fail. Algorithms are presented and analyzed that allow the conflicting stations to compute a stochastic estimate <italic>n</italic><supscrpt>*</supscrpt> of <italic>n</italic>, cooperatively, at small cost, as a function of the feedback elicited during its execution. An algorithm to <italic>resolve</italic> a conflict among two or more stations controls the retransmissions of the conflicting stations so that each eventually transmits singly to the channel. Combining one of our estimation algorithms with a tree algorithm (of Capetanakis, Hayes, and Tsybakov and Mikhailov) then leads to a hybrid algorithm for conflict resolution. Several efficient combinations are possible, the most efficient of which resolves conflicts about 20 percent faster on average than any of the comparable algorithms reported to date.

Computational techniques for accurate performance evaluation of multirate, multihop communication networks

Computational techniques are presented for the connection-level performance evaluation of communication networks, with stochastic multirate traffic, state-dependent admission control, alternate routing, and general topology-all characteristics of emerging integrated service networks. The techniques involve solutions of systems of fixed-point equations, which estimate equilibrium network behaviour. Although similar techniques have been applied with success to single-rate fully connected networks, the curse of dimensionality arises when the techniques are extended to multirate, multihop networks, and the cost of solving the fixed point equations exactly is exponential. This exponential barrier is skirted by exploiting, in particular, a close relationship with the network reliability problem, and by borrowing effective heuristics from the reliability domain. A series of experiments are reported on, comparing the estimates from the new techniques to the results of discrete-event simulations.

A lower bound for probabilistic algorithms for finite state machines

Abstract Freivalds recently reported a construction of a 2-way probabilistic finite automaton M that recognizes the set { a m b m : m ⩾ 1} with arbitrarily small probability of error. This result implies that probabilistic machines of this type are more powerful than their deterministic, nondeterministic, and alternating counterparts. Freivalds construction has a negative feature: the automaton M runs in Ω(2 n 2 n) expected time in the worst case on inputs of length n . We show that it is impossible to do significantly better. Specifically, no 2-way probabilistic finite automaton that runs in exp( o ( n )) expected time recognizes { a m b m : m ⩾ 1} with probability of error bounded away from 1 2 . In passing we derive results on the densities of regular sets, the fine structure of Freivalds construction, and the behavior of random walks controlled by Markov chains.