Uri Yechiali

Scheduling deteriorating jobs on a single processor

N jobs are to be processed sequentially on a single machine. While waiting for processing, jobs deteriorate, causing the random processing requirement of each job to grow at a job-specific rate. Under such conditions, the actual processing times of the jobs are no longer exchangeable random variables and the expected makespan is no longer invariant under any scheduling strategy that disallows idleness. In this paper, we analyze the effects of different deterioration schemes and derive optimal scheduling policies that minimize the expected makespan, and, for some models, policies that minimize the variance of the makespan. We also allow for random setup and detaching times. Applications to optimal inventory issuing policies are discussed and extensions are considered.

Analysis of customers' impatience in queues with server vacations

AbstractMany models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival. In this paper we consider systems with servers vacations where customers’ impatience is due to an absentee of servers upon arrival. Such a model, representing frequent behavior by waiting customers in service systems, has never been treated before in the literature. We present a comprehensive analysis of the single-server, M/M/1 and M/G/1 queues, as well as of the multi-server M/M/c queue, for both the multiple and the single-vacation cases, and obtain various closed-form results. In particular, we show that the proportion of customer abandonments under the single-vacation regime is smaller than that under the multiple-vacation discipline.

Dynamic priority rules for cyclic-type queues

A cyclic service system is composed of K channels (queues) and a single cyclically roving server who typically takes a positive amount of time to switch between channels. Research has previously focused on evaluating and computing performance measures (notably, waiting times) of fixed template routing schemes under three main service disciplines, the exhaustive, gated and limited service regimes. In this paper, probabilistic results are derived that allow control strategies and optimal policies to be considered for the first time. By concentrating on a new objective function, we are able to derive rules of index form amenable for direct implementation to dynamically control the system at suitably defined decision epochs. These rules utilize current system information, are of an adaptive nature, and are shown to emanate from a general physical principle. CYCLIC QUEUES; POLLING SYSTEMS; STOCHASTIC SCHEDULING

Cyclic reservation schemes for efficient operation of multiple-queue single-server systems

We study two new cyclic reservation schemes for the efficient operation of systems consisting of a single server and multiple queues. The schemes are the Globally Gated regime and the Cyclic-Reservation Multiple-Access (CRMA). Both procedures possess mechanisms for prioritizing the queues and lend themselves to a closed-form analysis. The combination of these two properties allows for effective and efficient operation of the systems, for which we provide a thorough delay analysis and derive simple rules for optimal operation.

An M/M/s Queue With Servers’ Vacations

AbstractIn an M/M/s queueing system a server that completes service and finds no waiting units in line leaves for a vacation of an exponentially distributed duration. At the end of the vacation the server returns to the main system. Two models are analysed. In the first, a server returning to an empty queue takes immediately another vacation. In the second, only a single vacation is taken each time. For model 1, formulas for the distribution of the number of busy servers and the mean number of units in system, L, are derived. Numerical calculations indicate that L is very closely a linear function of the mean vacation time. Finally it is shown that model 2 may be analysed similarly to model 1.

Queues with system disasters and impatient customers when system is down

Abstract Consider a system operating as an M/M/c queue, where c=1, 1<c<∞, or c=∞. The system as a whole suffers occasionally a disastrous breakdown, upon which all present customers (waiting and served) are cleared from the system and lost. A repair process then starts immediately. When the system is down, inoperative, and undergoing a repair process, new arrivals become impatient: each individual customer, upon arrival, activates a random-duration timer. If the timer expires before the system is repaired, the customer abandons the queue never to return. We analyze this model and derive various quality of service measures: mean sojourn time of a served customer; proportion of customers served; rate of lost customers due to disasters; and rate of abandonments due to impatience.

Queues with slow servers and impatient customers

Abstract We study M / M / c queues ( c = 1 , 1 c ∞ and c = ∞ ) in a 2-phase (fast and slow) Markovian random environment, with impatient customers. The system resides in the fast phase (phase 1) an exponentially distributed random time with parameter η and the arrival and service rates are λ and μ, respectively. The corresponding parameters for the slow phase (phase 0) are γ, λ 0 , and μ 0 ( ⩽ μ ) . When in the slow phase, customers become impatient. That is, each customer, upon arrival, activates an individual timer, exponentially distributed with parameter ξ. If the system does not change its environment from 0 to 1 before the customer’s timer expires, the customer abandons the queue never to return. We concentrate on deriving analytic solutions to the queue-length distributions. We derive, for each case of c, the corresponding probability generating function, and calculate the mean queue size. Several extreme cases are investigated and numerical results are presented.

Analysis and control of polling systems

We present methods for analyzing continuous-time multi-channel queueing systems with Gated, Exhaustive, or Globally-Gated service regimes, and with Cyclic, Hamiltonian or Elevator-type polling mechanisms. We discuss issues of dynamically controlling the servers order of visits to the channels, and derive easily implementable index-type rules that optimize systems performance. Future directions of research are indicated.

Waiting Times in the Non-Preemptive Priority M/M/c Queue

By using a probabilistic equivalence between the M/G/1 queue with multiple server ’ vacations and the M/M/c system, we derive the Laplace-Stieltjes transform of the waiting time Wk, of a class-k customer in the non-preemptive priority M/M/c queue where all customers have the same mean service time. We also calculate the first two moments of Wk.

Infinite-server queues with system's additional tasks and impatient customers

A system is operating as an M/M/∞ queue. However, when it becomes empty, it is assigned to perform another task, the duration U of which is random. Customers arriving while the system is unavailable for service (i.e., occupied with a U-task) become impatient: Each individual activates an “impatience timer” having random duration T such that if the system does not become available by the time the timer expires, the customer leaves the system never to return. When the system completes a U-task and there are waiting customers, each one is taken immediately into service. We analyze both multiple and single U-task scenarios and consider both exponentially and generally distributed task and impatience times. We derive the (partial) probability generating functions of the number of customers present when the system is occupied with a U-task as well as when it acts as an M/M/∞ queue and we obtain explicit expressions for the corresponding mean queue sizes. We further calculate the mean length of a busy period, the mean cycle time, and the quality of service measure: proportion of customers being served.