Abstract: A special kind of priority queue structure, priority trees (p-trees), and an algorithm for building such trees are investigated. The main part of the paper is devoted to a mathematical analysis of the algorithm showing that the expected number of key comparisons to insert a “random” element into a “random”p-tree withn nodes isO((logn)2). Also some practical experiments, comparing different types of priority queue strategies, are presented.

Abstract: This paper studies a controlled queueing system in which the decision- maker may change servers according to rules which depend only on the queue length. It is proved that for a given control policy a properly normalised sequence of these controlled queue length processes converges weakly to a controlled diffusion process as the queueing systems approach a state of heavy traffic. This paper considers controlled queueing systems. We use the term control- led queue to describe a queueing process in which a decision maker has the ability to change the parameters of the system as he observes the state of the system. In particular, we will consider controlled queues where the decision- maker may change servers according to rules which depend only on the queue length. The paper by Prabhu and Stidham (9) is a recent survey of the problem of controlling queues in which they discuss and suggest the need for new approaches. Whitt (15) describes the work that has been done studying queues in heavy traffic via weak convergence theory. This paper is an attempt to bring these two fields together and provide a basis for useful approximations for controlled queues without distributional assumptions. We extend results of Iglehart and Whitt (7) for the weak convergence of a sequence of queues in heavy traffic to a diffusion process to the weak convergence of a sequence of controlled queues in heavy traffic to a controlled diffusion. We say that a controlled queue is in heavy traffic if the mean service rate for each available server is approximately equal to the mean arrival rate. It is then possible to study control policies for the queueing models by studying policies for the .diffusion process, a more tractable problem although still difficult. It is shown in (10) that the optimal policy for the limiting diffusion process under an average

Abstract: For an M/G/1 queuing system, this paper assumes (l) that the discipline is administratively constrained to a “head-of-the-line” rule with n classes, and (2) that priorities are centrally assigned to customers according to their individual cost per unit time spent in the system, or their individual service-time requirements, or both (as the available information may be), but independently of the state of the queue. Then optimal priority-assignment rules for the various information structures are characterized. When n = 2, the higher priority may always be optimally assigned to all customers who are “no worse than average” and with sufficiently high load to “practically all” customers.

Abstract: A generalized multi-entrance and multipriority M/G/1 time-sharing system is dealt with. The system maintains many separate queues, each identified by two integers, the priority level and the entry level The arrival process of users is a homogenous Poisson process, while service requirements are identically distributed and have a finite second moment. Upon arrival a user joins one of the levels, through the entry queue of this level. In the (n, k)-th queue, where n is the priority level and k is the entry level, a user is eligible to a (finite or infinite) quantum of service. If the service requirements of the user are satisfied during the quantum, the user departs, and otherwise he is trans- ferred to the end of the (n + 1, k)-th queue for additional service. When a quantum of service is completed, the highest priority nonempty level is chosen to be served next; within this level the queues are scanned according to the priority of their entry level, and the user at the head of the highest priority nonempty queue is chosen to be served. In such a priority discipline, preferred users always get an improved service, though the service of all users is degraded in proportion to their service requirements. Expected flow times and expected number of waiting users are derived and then specialized to the head-of-the-line M/G/1 priority discipline (in which quanta have infinite length and service is uninterrupted) and to the FBn time-sharing system. Finally, the generalized multientrance and multipriority time-sharing discipline is (numerically) compared with several other time-sharing systems.

Abstract: The load carried by a queuing system under equilibrium conditions is the average amount of server usage per unit of time. In telephony, this parameter is often evaluated by recording the number of busy servers at regular time intervals; these readings are then cumulated and their sum, after division by the number of observations, is an unbiased estimate of the carried load. The purpose of this paper is to derive exact formulas for the computation of the variance of this measurement in systems with arbitrary input and departure rates. The results obtained here thus apply to a wide class of teletraffic models which includes, in particular, the delay-and-loss systems with finite- or infinite-source inputs, exponential service times, and arbitrary defection rates from the queue. Problems related to computations are also considered, special attention being paid to the reduction of both computer time and storage when the number of states is large.

Abstract: This paper presents formulas for calculating waiting time for customers in a queue with combined preemptive and head-of-line (nonpreemptive) priority scheduling disciplines and describes the reasoning behind them. This work has been applied in the development of programmable terminal control units.

Abstract: This paper consider the probability distribution of the busy period of a single server queuing system with constant servicing time for each customer when customer arrive condonly in varying batch sizes and the batch size is distributed according to the Poisson law. First four moments and a few limiting forms of the distribution are also given.

Abstract: This paper considers a multiserver queue with heterogeneous servers in which arriving customers are assigned to servers according to an autonomous Markov chain. Under conditions ensuring stability, stationary distributions are computed for the waiting times and the system response times of successive customers. Limits are also obtained for the expected system workload and the expected number of customers in the system.

Abstract: The paper presents a detailed study of a computer-controlled queuing system with Poisson input, first-come first served queue discipline, multiple exponential servers, switching network and blocked customers leave the system permanently without services. A generating function for the stationary-state probabilities of the system and a necessary and sufficient condition for the existence of statistical equilibrium for the system are derived. Further, a formula for the determination of the waiting-time distribution in the system is obtained by the employment of the theory of Markov chains.

Abstract: The writer believes that unique interpretation of the results in the subject paper [ I ] is likely only under severely modified assumptions. The problem formulation by Chang centers on ' 6. .. a multi-server system with c servers. \" In the absence of details regarding the service discipline, there is a temptation to assume that immediate service is given each new arrival if one or more of the c servers is idle. Indeed, Eqs. (2) and (3) [ 1, p. 3701 are recognized from the literature [2, p. 3781 as being compatible with such a server discipline. Difficulty arises, however, if definition (4) [ 1, p. 37 13 for transition probabilities is interpreted in that context. That is, the indicated discipline for immediate service implies that the number of busy servers, i, may not be constant during the service time for any particular item. Then, the notation P i (j) , where j is the number of new arrivals during a service time x, is clearly unsatisfactory for denoting transition probabilities. However , the importance of (4) to all subsequent developments in the paper is clear from the mathematics, and deference is now accorded to (4). A highly motivated reader might attempt to devise a hypothetical system which is oriented directly to (4) and (5) [ 1, p. 3711. H e could begin by assuming that his system had two units: a temporary storage buffer for new arrivals and a service unit which could accept up to c items at a time from the buffer. Now the buffer could be polled at the end of every service unit action. T o accom-424 E. W. STACY modate the first term on the right-hand side of each equation in (5) , the buffer might also be polled at later times whenever it is empty at a particular polling time. Here an indirect contrast is highlighted for multiserver vs batch-service queuing systems. If the latter is intended, mystery envelops the assumption [ 1, p. 3701 that \". .. processing time is closer to being constant than exponentially distributed. \" Definition of the word 'closer' is not given in the paper except possibly via inference from the numerical example. Two Erlang-2 distributions, with parameters 0.25 and 0.20, are stipulated in that example. The first of these yields a mean service time of 8 and about a 35 percent probability for service time that …

Abstract: The paper presents an investigation of a computer-controlled queuing system with a switching network, feedback and where blocked customers may become first in the queue, rejoin the end of the queue or leave the system permanently without service A generating function for stationary-state probabilities of, the necessary and sufficient condition for the existence of statistical equilibrium for, and a formula for the determination of the waiting-time distribution in the system are derived

Abstract: The paper presents a detailed study of a computer-controlled queuing system with Poisson input, first-come first served queue discipline, multiple exponential severs and general access-cycle times. A generating function for the state probabilities of, and the necessary and sufficient condition for the existence of, statistical equilibrium for the system are obtained. Further, the average number of customers in the system is derived from the generating function.

Abstract: The behaviour in equilibrium of networks of queues in which customers may be of different types is studied. The type of a customer is allowed to influence his choice of path through the network and, under certain conditions, his service time distribution at each queue. The model assumed will usually cause each service time distribution to be of a form related to the negative exponential distribution. Theorems 1 and 2 establish the equilibrium distribution for the basic model in the closed and open cases; in the open case the individual queues are independent in equilibrium. In Section 4 similar results are obtained for other models, models which include processes better described as networks of colonies or as networks of stacks. In Section 5 the effect of time reversal upon certain processes is used to obtain further information about the equilibrium behaviour of those processes.

Abstract: We consider a single-server queuing system with several classes of customers who arrive according to independent Poisson processes. The service time distributions are arbitrary, and we assume a linear cost structure. The problem is to decide, at the completion of each service and given the state of the system, which class (if any) to admit next into service. The objective is to maximize the expected net present value of service rewards received minus holding costs incurred over an infinite planning horizon, the interest rate being positive. One very special type of scheduling rule, called a modified static policy, simply enforces a (nonpreemptive) priority ranking except that certain classes are never served. It is shown that there is a modified static policy that is optimal, and a simple algorithm for its computation is presented.

Abstract: We consider a nonpreemptive priority queue with a finite number of priority classes, Poisson arrival processes, and general service time distributions. It is not required that the system be stable or even that the mean service times be finite. The economic framework is linear, consisting of a holding cost per unit time and fixed service reward for each customer class. Future costs and rewards are continuously discounted with a positive interest rate. Allowing general initial queue sizes, we develop an expression for the expected present value of rewards received minus costs incurred over an infinite horizon. From this we obtain the Laplace transform of the time-dependent expected queue length for each customer class.