Zvi Rosberg

Optimal control of service in tandem queues

Customers arrive in a Poisson stream into a network consisting of two M/M/1 service stations in tandem. The service rate u \in [0, a] at station 1 is to be selected as a function of the state ( x_{1}, x_{2} ) where x i is the number of customers at station i so as to minimize the expected total discounted or average cost corresponding to the instantaneous cost c_{1}x_{1} + c_{2}x_{2} . The optimal policy is of the form u=a or u=0 according as x_{1} and S is a switching function. For the case of discounted cost, the optimal process can be nonergodic, but it is ergodic for the case of average cost.

Packet delay under the golden ratio weighted TDM policy in a multiple-access channel

Consider n transmission stations sharing a single communication channel. Packets arrive at the stations according to n independent renewal processes, possibly with different rates. The transmitters are assumed to be able to store an unlimited number of packets in their buffers. The stations transmit packets during time slots allocated to them according to a given {\em conflict-free distributed protocol.} The cost criterion according to which protocols are evaluated is the long-run weighted average buffer occupancies. (The average waiting time is a special case of such a weighting.) A lower bound to the cost criterion under time division multiplexing (TDM) protocols is given, and the costs of two protocols are analyzed. The first protocol is the {\em random-control} policy, and the second is the {\em golden ratio} policy which is shown to achieve a cost close to the lower bound for realistic parameters.

Optimal routing to parallel heterogeneous servers-small arrival rates

Consider a set of k(>or=2) heterogeneous and exponential servers that operate in parallel. Customers arrive into a single infinite capacity buffer according to a Poisson process, and are routed to available servers in accordance with some routing policy. It is shown that for arrival rates in some possible interval (0, lambda /sub 0/), ever routing policy which minimizes the long-run expected holding cost is contained in the set of routing policies that minimize the expected flow time for a system with fixed initial population and no new arrivals. >

TDM policies in multistation packet radio networks

A multistation packet radio network with m stations and a finite number of nodes n that uses a conflict-free protocol to access the backbone network of stations through a shared channel is discussed. The goal is to derive an allocation of the channel time slots (time-division multiplexing cycle), so that all transmissions will be conflict-free and some measure of performance (e.g., the expected total weighted throughput, the expected weighted holding cost) will be optimized. The methodology that is used is to bound the performance and to allocate the slots according to the golden ratio policy. >

Optimal decentralized control in a multiaccess channel with partial information

We consider two transmission stations sharing a single communication channel. For different values of the input message rates r_{i}, i = 1,2 , a simple open-loop control policy is shown to be optimal for the long-run average throughput criterion.

Deterministic Routing to Buffered Channels

Consider n exponential transmission channels which transmit information with different rates. Every channel has a buffer which is capable of storing an unlimited number of messages. A new message first arrives at the controller, which immediately routes it to one of the channels according to an infinite deterministic routing sequence. A cost per unit of staying time is charged in each of the channels (channel dependent cost), and the long-run average staying cost is taken as the cost criterion. For every n and a Poisson arrival process, a lower bound to the cost is found and a new routing policy, the golden ratio policy, is presented and its cost is evaluated. It is shown that for a variety of system parameters, the golden ratio routing policy has a cost close to the lower bound.

Customer routing to parallel servers with different rates

We consider the problem of routing customers to parallel servers having different rates. There are no buffers in the system. Each customer must be rooted to a server immediately upon its arrival and if the server to which it is routed is occupied, then the customer is aborted. The aim is to maximize throughput (the proportion of customers which are successfully routed to a free server), when the routing must be done without knowing which servers are occupied and which are free. An upper bound on the throughput is found for a general renewal arrival process and geometric service times. Furthermore, a new routing policy, the golden ratio policy, is suggested and shown to approach a limit which is within at least 98.4 percent of the upper bound. The golden ratio policy is a generalization of the round robin policy, when the service rates of the servers are different.

Customer scheduling under queueing constraints

A scheduling problem of an exponential single server with a finite queueing capacity that serves customers from n heterogeneous classes is considered. Arrivals are Poissonian and every class has its own rate and its own finite waiting room. The waiting rooms can be of arbitrary size. Arriving customers that find a full queue are lost. Of particular interest is finding a scheduling policy that allows service preemption and has a weighted throughput which is close enough to the optimal one. As an optimal scheduling policy is extremely hard to find, a different methodology is used to tackle the problem. First, the optimal weighted throughput is bound from above, and the asymptotically optimal policy is found. Then, based on the bounding technique and the asymptotically optimal policy, a new policy, the overflow scheduling policy, that provides a weighted throughput which is very close to the upper bound is proposed. The quality of the policy is demonstrated by various examples. >

A hybrid access method in a multiple access channel

Abstract This paper considers a multiple-access communication channel with an infinite number of users. We show that if a controlled slotted Aloha protocol is used, then messages with variable length will have a negative impact on the average message delay. In order to alleviate this problem, a mixed mode ( Hybrid ) access method is suggested under which the channel bandwidth is split into two sub-channels, managed under different policies. Messages whose length is less than or equal to a critical value are transmitted in one sub-channel under a slotted Aloha policy. The rest of the messages are sent through a separate portion of the channel bandwidth, using a reservation protocol. We show that under this hybrid access method the average delay of a message is greatly improved.

Bounds for the optimal decentralized access protocol in a local area network

Abstract Here, we consider a local area network under the set of decentralized access protocols. Lower and upper bounds are obtained for the optimal long-run average waiting cost. These bounds are shown to be quite tight for a typical network.