Queues with moving average service times

Expressions are derived for the mean response times of each priority level in a multiserver M/M/m queue operating under preemptive resume scheduling. Exact results are obtained for cases where all priorities have the same mean service times; approximate results are obtained for the more general case where mean service times may differ. The results hold for any number of servers and any number of classes. For each priority level, it is assumed that arrivals are Poisson and service times are exponentially distributed.

It is well known that queues with exponentially distributed service times have the smallest Shannon capacity among all single-server queues with the same service rate. In this paper, we study the capacity of timing channels in which the service time distributions have bounded support, i.e., Bounded Service Timing Channels (BSTC). We derive an upper bound and two lower bounds on the capacity of such timing channels. The tightness of these bounds is investigated analytically as well as via simulations. We find that the uniform BSTC serves a role for BSTCs that is similar to what the exponential service timing channel does for the case of timing channels with unbounded service time distributions. That is, when the length of the support interval is small, the uniform BSTC has the smallest capacity among all BSTCs.

We propose a static service differentiation policy for a single-server queueing system serving homogeneous customers. We show that by randomly assigning customers different service grades with different service rates, the average waiting time can be reduced without affecting the mean service time. Such differentiation introduces more service time variability, but it also creates information that enables the implementation of service rate-based scheduling, which mitigates the increased variance and may even reduce the total waiting time. We provide conditions under which our static service differentiation reduces waiting, and further derive closed-form expressions for the optimal differentiation policy, which shows that both optimal service rates and allocation probabilities form geometric sequences. We illustrate our policy in the context of quality-based service domains, in which customers value service time but dislike waiting. Numerically, we find that providing differentiated service can improve system performance by 5% without any additional capacity.

We have conducted a research in order to clarify a movement of a top table tennis player using DLT method. We have tried to obtain basic data such as time intervals of various playing, times of shots, playing area, distance of movement and so on to plan a training program of a movement for table tennis.The objects of analysis were 5 games of YOO Nam Kyu, who was the SEOUL Olympic champion, in the 9th ASIAN Table Tennis Championships in Japan. The games were filmed at 60 frames/second with 6 VTR cameras which were installed at separate locations in the gymnasium. For the purpose of this analysis, the various time intervals are defined as follows: In-play Time was from the initial movement of the service to final contact with the ball during play. It is acknowledged that this definition differs slightly from the official ITTF regurations for “ball in play” time. Rally Time was from contact with the ball on service to final contact with the ball during play. Service Time was from the initial movement of the service to contact with the ball on service. Waiting Time was from one player's contact with the ball to the other player's contact.Two-dimensional coordinates on a TV monitor were computed with a VTR motion analyzer connected to a personal computer. These coordinates were then converted to real (X-Y-Z) coordinates using DLT method. The position of the player was estimated from the coordinates of the player's head (vertex), center of shoulders (acromions), center of hips (trochanterions) and center of ankles (malleolus fibulae points). Distance of movement and velocity of movement were computed for every body point in several directions; right to left and forward to backward (in relation to the table), and up to down. Distance of movement was a line that links the position of the player every 0.1 sec during Rally-time. Velocity of movement was calculated by dividing distance of movement by Rally-time for each rally.The results and findings are summarized as follows;1) Mean total In-play time was 144.63±27.69 sec (±SD). This represented about 20-30% of the games of which average was 611.75±33.50 sec long, meaning that about 70-80% of the games was excess time spent doing such things as picking up the ball or wiping off perspiration.2) Mean Service Time, Mean Rally Time, Mean In-play Time and Mean Waiting Time was 1.05±0.37 sec, 2.56±1.79 sec, 3.60±1.83 sec and 0.66±0.19 sec, respectively. This represented about 30% of In-play Time was Service Time and about 70% of it was Rally Time.3) Mean times of shots during 1 rally was 4.61±2.12. The distribution of times of shots during 1 rally showed that in the case of 74.6% of all rally, a player won or lost a point in 5 shots.4) The playing area for YOO was positioned at the side of his free hand in 4 games. In 1 game, it was positioned at the side of his racket hand. His main stroke of the game was different from previous 4 games.5) In each game, the distance of movement for YOO on a horizontal plane during Rally Time was between about 100m and 200m at the center of the ankles. And mean distance of movement for YOO on a horizontal plane during 1 rally was between about 3.5m and 4.0m at the center of the ankles. In the table tennis games, the player's repeated acceleration and deceleration in various directions was instantaneous.6) In each game, mean average velocity of movement for YOO on a horizontal plane during 1 rally was between 0.85m/sec and 1.08m/sec at the head, 0.90m/sec and 1.05m/sec at the center of the shoulders, 0.99m/sec and 1.18m/sec at the center of the hips and 1.36m/sec and 1.51m/sec at the center of the ankles.7) It seems that these new data and findings are very important to plan an efficient training program for table tennis coaches and players.

Robust bounds and throughput guarantees for closed multiclass queueing networks

Given the distribution of natural losses from any rank of a graded manpower system, and the rate of promotion to higher ranks, we define the mobility of rank members as a function of the cumulative lengths of service of all rank members and those members destined for promotion. We show that when the rank contains many individuals, and promotions are made without regard to seniority, mobility decreases with time in service if the natural loss function has an increasing failure rate, and increases with time in service if the loss function has a decreasing failure rate; mobility increases with time in service if promotions are awarded by seniority. We also derive bounds for the average time in service of rank members, based on the loss function and the promotion rate. In addition, we compare the average time to promotion, average time to resignation, and average time in service for the two promotion policies mentioned above. Our results show that promotion by seniority produces the longest possible average member service times if the loss function has an increasing failure rate, and the shortest average service times if the loss function has a decreasing failure rate. Using real data, we derive and categorize the theoretical natural loss distribution for a system of university faculty, the implied promotion policy governing the transition-to-tenure process, and the individual mobility function for nontenured members. This analysis does not require a large amount of data.

Power interruptions cause losses, for consumers and for the utility, but it is necessary in order to make improvements in the power distribution system and also for protective devices operate when an outage occurs. Therefore, it is important that utilities perform the service for emergency occurrences as quickly as possible, reducing the time of the outage. Brazilian National Electric Energy Agency (ANEEL) requires the utilities send every month their average service time, regulating the quality of the service, and determines the maximum length of interruptions for each set of Consumer Units and also their maximum quantity of occurrences. Therefore, reducing measures in service time is a great interest of utilities. The installation of smart meters allows that the distribution company be informed of the interruption at the same time it occurs, also determining the fault location, everything because of two-way flow of information between the meter and the control center. Thus, the time of fault location is reduced, resulting in reducing the time of the occurrence service and total interruption time for those consumers achieved. Thus, there is impact on the system average interruption duration index (SAIDI), increasing system reliability. In addition to this gain, there is a reduction of expected energy not supplied (EENS). In this context, the evaluation of the SAIDI and EENS reduction because of the replacement of conventional meters with smart meters.

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (kappa, K) for the queue length distribution in the general case. Moreover, we prove that the order m^2 of the distribution cannot be further reduced in general. Examples for which the order is between m and m^2 are also identified. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (kappa, K). Moreover, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues

Call centres are becoming increasingly important in our modern commerce. We are interested in modelling the time-varying pattern of average customer service times at a bank call centre. Understanding such a pattern is essential for efficient operation of a call centre. The call service times are shown to be lognormally distributed. Motivated by this observation and the important application, we propose a new method for inference about non-parametric regression curves when the errors are lognormally distributed. Estimates and pointwise confidence bands are developed. The method builds upon the special relationship between the lognormal distribution and the normal distribution, and improves upon a naive estimation procedure that ignores this distributional structure. Our approach includes local non-parametric estimation for both the mean function and the heteroscedastic variance function of the logged data, and uses local polynomial regression as a fitting tool. A simulation study is performed to illustrate the method. We then apply the method to model the time-varying patterns of mean service times for different types of customer calls. Several operationally interesting findings are obtained and discussed. Copyright © 2006 John Wiley & Sons, Ltd.

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process. Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue. In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

Conventional wisdom suggests that a fast single-server queue is preferable to a number of slower single-server queues. The rationale is that if the service rate of the superposed queue is the sum of the service rates of the individual queues, then the average waiting time for the superposed queue is smaller than the weighted average waiting time of the individual queues. This argument, however, assumes that the service time of the customers is perfectly scalable. Unfortunately, this assumption rarely holds in reality. In this paper we analyze the superposition of M/G/1 queues. We assume that there are scalable and unscalable components in the service time. On the basis of the criteria of average waiting time and average response time, we derive necessary and sufficient conditions for whether the superposition of queues is preferable. Examples of magnetic rotational disks are given to illustrate our findings.

Conventional wisdom suggests that a fast single-server queue is preferable to a number of slower single-server queues. The rationale is that if the service rate of the superposed queue is the sum of the service rates of the individual queues, then the average waiting time for the superposed queue is smaller than the weighted average waiting time of the individual queues. This argument, however, assumes that the service time of the customers is perfectly scalable. Unfortunately, this assumption rarely holds in reality. In this paper we analyze the superposition of M/G/1 queues. We assume that there are scalable and unscalable components in the service time. On the basis of the criteria of average waiting time and average response time, we derive necessary and sufficient conditions for whether the superposition of queues is preferable. Examples of magnetic rotational disks are given to illustrate our findings.

Some statistical analysis of design data.- Line service times and utilization.- First concepts and relations of queuing theory.- A first resuot on the single-server queue.- A general result on the single-server queue.- Termina systems with polling.- Some results on queues with priority disciplines.- Some applications of Markov processes.- Core buffer requirements.- Service times of direct-access storage devices.- Queuing for data accesses.- Final remarks on the computer center analysis.- Imbedded Markov chains of some queuing processess.- Limit utilization and wait time distribution.

In this paper, a single-server queue with non-homogeneous Poisson input and general service time is considered. Particular attention is given to the case where the parameter of the Poisson input λ(t) is a periodic function of the time. The approach is an extension of the work of Takács and Reich . The main result of the investigation is that under certain conditions on the distribution of the service time, the form of the function λ(t) and the distribution of the waiting time at t = 0, the probability of a server being idle P 0 and the Laplace transform Ω of the waiting time are both asymptotically periodic in t. Putting where b(t) is a periodic function of time, it is shown that both P o and Ω can be expanded in a power series in z, and a method for calculating explicitly the asymptotic values of the leading terms is obtained. In many practical queueing problems, it is expected that the probability of arrivals will vary periodically. For example, in restaurants or at servicestations arrivals are more probable at rush hours than at slack periods, and rush hours are repeated day after day