Showing papers on "Queue management system published in 1971"

Characterization and Computation of Optimal Policies for Operating an M/G/1 Queuing System with Removable Server

Abstract: This paper studies the optimal operation of an M/G/1 queuing system with removable server and the following cost structure: a holding cost per customer in the system per unit time, a cost per unit time of keeping the server running, and fixed charges for turning the server on or off. The server can be turned on at arrival epochs or off at service-completion epochs. The paper characterizes an optimal policy for the infinite-horizon discounted problem, offers an optimality proof, and presents a computational algorithm.

Analysis of system bottlenecks using a queueing network model

Abstract: While it is well known that queues can build up at various points in large scale multiprogramming computer systems, comparatively little is known about the factors which govern the lengths of these queues and their relationship to overall system performance. The first part of this paper is concerned with developing a queueing network model which can be used to study a number of such questions. The model is then applied to a specific problem concerning the proportion of processing requests which should be directed to each of a set of functionally equivalent peripheral processors (e.g., disks and drums) in order to optimize overall system performance. A surprising result is that optimal performance is not attained when queue lengths and processor utilization percentages are equal, but rather when the fastest processor has the longest expected queue and is in effect creating a system bottleneck.

Two Queues with Changeover Times

Abstract: A single server attends to two separate queues. Each queue has Poisson arrivals and a general service time distribution. A changeover time, with a general distribution, is required whenever the server crosses from one queue to the other. This paper investigates two queue disciplines: alternating priority and strict priority. In each case, it obtains the Laplace-Stieltjes transforms of the waiting-time distributions for a stationary process, as well as the first moments of the waiting-time distributions.

The Remaining Busy Period of a Finite Queue

Abstract: In the M/G/1/N queuing system where no more than N < ∞ customers are allowed in the system, define the remaining busy period from state i = 0, 1, …, N as the time spent in transition from state i to state 0, i.e., until the server becomes idle for the first time. The purpose of this paper is to derive the Laplace-Stieltjes transform of the distribution of this time and the generating function of the number of customers served in this time.

A Dynamic Time-Sharing Priority Queue

Abstract: This paper deals wi th a single server s t a t ion del ivering service to m pr ior i ty classes (m might be finite or infinite). The arr ival process of customers from a j t h (j = 1, 2 , . . . , m) pr ior i ty class ( j t h customers) to a single server s t a t ion is a homogenous Poisson process wi th ra te X3 • Service requi rements of j t h customers are exponent ia l ly d i s t r ibu ted wi th mean 1/~. The wait ing line consists of infinitely many separate queues all of which obey the F I F O rule. Each pr ior i ty class is assigned to one of the queues. A newly arr ived j t h customer joins the end of a prede termined queue. In the i t h (i = 1, 2 , . . . ) queue a customer is eligible for a cer ta in amount of service following which he e i ther depar ts when his service requi rement has been satisfied or joins the end of the (i + 1) th queue for addi t ional service. When a q u a n t u m of service is completed, the server a t t ends to the first customer in the lowest index (highest pr ior i ty) nonempty queue. In such a pr ior i ty regime, long and unknown in advance service requi rements in all pr ior i ty classes are dynamical ly penalized by degrading the i r pr ior i ty degree. This paper derives ma themat i ca l expressions for calcula t ing the expected to ta l flow t ime of a j t h customer whose service requ i rement is known. K E Y W O R D S A N D P H R A S E S : t ime-shar ing, operat ive systems, computers , queues, s tochast ic processes, ma themat i ca l s ta t is t ics , ma themat i ca l models CR CATEGORIES: 3.89, 4.39, 5.5 Introduction One of the main features characterizing time-sharing disciplines is the priority given to jobs with short and unknown in advance requirements, at the expense of jobs with long ones. In the simplest time-sharing discipline a customer admitted for service is eligible to stay in service for no longer than a predetermined time interval. When the customers' requirement has not been completed during this time interval, he is dismissed and placed at the end of the queue, and the next customer admitted to service is the first in the queue. In due time when his turn comes up again, the same procedure is repeated until his requirement has been completed and he departs. A newly arrived customer joins the end of the queue. The length of time a customer stays uninterrupted in the service station is called a quantum. The discipline described above is known as Round-Robin (R.R.). The R.R. regime has been studied by Kleinrock [1], Adiri and Avi-Itzhak [2], and others [3]. One of the inherent weaknesses of an R.R. discipline is its ignorance of the amount of service (number of service quanta) already given to the customers present in the system. The priority given to short service demands might be improved by differentiating the customers present in the system according to the number of Copyr ight © 1971, Associat ion for Comput ing Machinery , Inc. General permission to republish, bu t not for profit, all or par t of this mater ia l is g ran ted provided t h a t reference is made to this publ ica t ion, to i ts da te of issue, and to the fact t h a t rep r in t ing privileges were gran ted by permission of the Association for Comput ing Machinery . * On leave of absence from the Techn ionI s rae l I n s t i t u t e of Technology. Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971, pp. 603-610. 6 0 4 I G A L A D I R I quanta they have already received. A customer who has received i service quanta will have priority over a customer that has received j quanta if i < j . The F.B.~ model is a proposed solution along the above lines. I t is comprised of one server and an infinite number of queues all of which obey the FIFO rule. A newly arrived customer joins the end of the first queue where, in due course, he receives a certain amount of service following which he either departs or joins the end of the second queue for additional service. The server, after completing a quantum, selects for service the first customer in the lowest index (highest priority) nonempty queue. This model was first suggested and studied by Schrage [4]. In the models discussed above all customers upon arrival are treated in the same way. In real life situations this is not always desirable and differentiation among certain groups of customers is preferred. The notion of improving the service of certain groups of customers at the expense of others is the basis of all priority disciplines. There are cases when it is desirable to couple the improvement of service to certain groups with discouragement to all jobs that have a long service requirement (the length of service requirement is not known in advance). This might be carried out by dynamically degrading the degree of a customer's priority according to the length of his service requirement. A generalized F.B.~ regime is proposed as a solution in accordance with the outlined features. A single server dispenses service to m independent infinitely large populations (m might be finite or infinite). Each of the m populations is assigned to one of the infinite queues in the F.B.~ model. Upon arrival a customer from the kth priority class joins the end of the appropriate queue. When his turn for service comes up he gains a nonpreemptive quantum of service. At the end of a quantum the next customer to be selected for service is the first customer in the lowest index (highest priority) nonempty queue. Upon completion of a service quantum in the j th ( j = 1, 2, • • • ) queue a customer either departs or joins the end of the ( j ~ 1)th queue; this might be viewed as diminishing by one his priority level. In a priority system working under such a regime long and unknown in advance, service demands in all priority classes are dynamically penalized by degrading their priority degree. This model was first suggested by Shemer [5], who has studied the case where no losses due to the generalized F.B.~ discipline are involved, however his results are in disagreement with ours. The Mathematical Model ASSUMPTIONS AND PRELIMINARY RESULTS. A single server dispenses service to an infinite number of priority classes. The arrival process of customers from the j th priority class ( j th customers) ( j = 1, 2, . . ,) is a homogeneous Poisson process with rate ks. Service requirements of j t h customers, L, are independent random variables identically exponentially distributed with mean 1/~. The waiting line consists of infinitely many separate queues. A newly arrived j th customer joins the end of the j th queue. When it is desired that upon arrival the j t h customers join the end of the ( j + k) th queue, we add k fictitious priority classes, namely, j, j + 1, j + 2, • • • , j -/k 1, with external arrival rates zero and the j th class becomes the ( j -~k) th class. In the case where the number of priority classes is finite, we can match priority classes with the appropriate queues and then substitute zeroes for all other external arrival rates. Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971 A Dynamic Time-Sharing Priority Queue 605 A f lh customer, upon admittance to service for the first time (in the f lh queue), is eligible for 0~ units of service time. If his service demand is shorter than 0j t ime units, he leaves the system the moment his requirement is satisfied, otherwise, after receiving 0j service time units, he joins the end of the (j --}1)th queue. In general, in the ith queue (i = 1, 2, . . ) a customer is eligible for 0~ units of service t ime during which he may complete his service and leave the system or, in the case where his demand is not satisfied, join the end of the (i W 1)th queue. Having completed a quantum at any one of the queues, the server admits the customer who is first in the lowest index nonempty queue. We assume that a quantum commences with a set-up time due to swapping and housekeeping activities. The quantum in the ith queue is comprised of two elements: a set-up time of length ri and a processing time not exceeding 0i units of time. Let us denote the length of a quantum in the kth queue by Qk • We have Qk £ ~Ok + rk with probabili ty ak = e \" O k /Dk with probabili ty 1 ak, k = 1, 2, . . . , (1) where d__ indicates equal in distribution and Dk has a density function fDk(X) = ~e-~(x-T~)/(1 -ak), rk _< X < 0~ -~ rk. (2) Equations (1) and (2) yield E(Qk) = rk--}(1 ak)/~, (3) and E(Qk 2) = 2(1 ak)/u 2 2 ( ( r k + 0k)o~k-r k ) / # + rk ~. (4) Deqning q~,k as the service time supplied to a customer in queues i, i + 1, • • • , k, given that the customer reaches the (/c + 1)th queue, and (~,k the probability of this event, we have { ~ (0r + r~) for k > i ql ,k = ~ ( 5 ) otherwise, and ~ i , k = ~ ( 6 ) otherwise. The arrival rate to the kth queue is k x~* = ~ ×~,,~_1. (7) i ~ 1 The proportion of time the server is busy delivering service in the kth queue is k pk = hk*E(Qk) = E(Qk) ~ k~.k_~. (8) i ~ 1 Hence, the proportion of time the station is busy supplying service to the first j queues is 1 i k P(i) = ~ p k = ~ E ( Q ~ ) ~ X,~,.k-1. (9) k = l k = l i = l Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971

Finite waiting space bulk service system

Abstract: This paper discusses the ergodic queue length distribution of a bulk service system with finite waiting space by the method of the imbedded Markov chain. The system under consideration is a queuing system with Poisson arrivals, general service times, single server and where service is performed on batches of random size.

Computer-controlled queuing systems with feedback

Abstract: A generalised approach is presented for the study of a computer-controlled queuing system with general input, ‘first-come-first-served’ queue discipline and multiple servers with feedback for either exponential servers or constant service times. Two generating functions for the state probabilities of the system corresponding to the two different types of service times are derived. Further, necessary and sufficient conditions for the existence of statistical equilibrium of the computer-controlled queuing system are obtained from the generating functions. The results obtained here generalise many results previously obtained by other workers.

Waiting-time distribution in computer-controlled queuing system

Abstract: The paper presents two formulas for the determination of the waiting-time distribution in a computer-controlled queuing system with Poisson input, multiple-exponential servers and first-come-first-served queue discipline. The formulas obtained are expressed in terms of the stationary-state probabilities of the computer-controlled queuing system. One of the formulas is derived by means of the formula of total proability and the other is derived by the use of the theory of Markov chains. It is shown that, by virtue of a limiting process, the first formula obtained in the paper yields the waiting-time distribution in the conventional queuing system without computer control. Furthermore, application of the two formulas for the determination of waiting-time distribution is illustrated by an example.

The effect of blocking on a single-server queue in tandem with a many-server system

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Performance evaluation of direct access storage devices with a fixed head per track

Abstract: Computer designers agree that the slow response of input and output devices is the most critical factor limiting the performance of a computer system. When a program requests an access to one of these devices, the request is put on a queue. The members of the queue are usually serviced on a first-come-first-served basis or on the basis of preassigned priorities. Devices with large capacity are able to store a number of data sets belonging to various programs in a multi-programming environment. This usually leads to a queue build-up for these devices, thereby creating a bottleneck in the system.

The Distribution of the Occupation Time for Single-Server Queues

Abstract: This paper gives a general method for finding the distribution of the occupation time for single-server queues, discusses in detail queues with recurrent input and general service times, and gives some examples in which either the Laplace-Stieltjes transform of the distribution function of the inter-arrival times or the Laplace-Stieltjes transform of the distribution function of the service times is a rational function.

Simulation of waiting line systems with queue length dependent parameters

Abstract: A queueing system is simulated in which the arrival and service rates are functions of the number of customers in the system. This type of system has been suggested as being more representative of real queueing situations than is the usual “simple” queue with constant arrival and service rates.A modification of the “time of next event” simulation method is required since an arrival to the queue changes the distribution of residual service time of the customer being served and the departure of a customer changes the distribution of the time of the next arrival.The primary purpose of the simulation is to examine the statistical problems involved in estimating the system parameters.