Single-server Case Research Articles

Appointment Scheduling for Multiple Servers

Appointment schedules, in essence, balance supply and demand and are often employed in settings where resources are scarce and thus a high utilization is realized (e.g., healthcare). Whereas most of the existing literature focuses on the single-server case, a framework is developed to study appointment scheduling in multiserver settings. Relying on phase-type approximations, general service-time distributions are modeled, which are fed into a recursive approach allowing evaluation and optimization of an objective function that balances expected waiting times and idle times. Studying optimized schedules for multiple servers reveals that the start and end of a session can deviate greatly from the dome-shaped pattern as established for the single-server case. Furthermore, a comparison of various multiserver setups shows that significant performance gains can be achieved when servers are pooled. This allows an explicit quantification of the cost of continuity of care. In addition, session overtime as well as early finish of servers can be incorporated in the approach; the benefits of the additional flexibility that a multiserver setting provides are summarized. For the stationary plateau of the dome, to which the optimal interarrival times converge, steady-state appointment schedules are obtained by exploiting the embedded Markov chain; these schedules are shown and argued to converge quickly to optimal solutions obtained in a heavy-traffic regime. In this regime, algebraic solutions are derived, which provide interesting managerial guidelines when the pooling of servers is considered in appointment scheduling. This paper was accepted by Bariş Ata, stochastic models and simulation.

Design and analysis of a fair, efficient and stable multi-server adaptive streaming protocol

Due to the surging bandwidth demand for applications like 360° video streaming, hypertext transfer protocol (HTTP) adaptive video streaming has been extended to fetch contents from a single server to multiple servers. However, when players share a bottleneck link, it fails to hold three critical properties: fairness, efficiency, and stability. Effective algorithms have been proposed to tackle the problem in the single-server case. We revisit the three properties and show that they degenerate in multi-server streaming. In this paper, we propose MultiFast, a protocol that offers the best user experience by achieving fair, efficient, and stable multi-server video streaming. MultiFast consists of (1) a bitrate selection algorithm that ensures fairness with concurrent connections; (2) a connection management algorithm that ensures efficiency and stability by managing connections based on network conditions. Our real-world experiments show that MultiFast achieves the best performance in fairness, efficiency, and stability metrics, outperforming its closest alternative.

Optimal Multiserver Scheduling with Unknown Job Sizes in Heavy Traffic

We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the singleserver k = 1 case, the optimal policy is the Gittins policy, but it is not known whether Gittins or any other policy is optimal in the multiserver case. Exactly analyzing the M/G/k under any scheduling policy is intractable, and Gittins is a particularly complicated policy that is hard to analyze even in the single-server case.

Optimal Multiserver Scheduling with Unknown Job Sizes in Heavy Traffic

We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the singleserver k = 1 case, the optimal policy is the Gittins policy, but it is not known whether Gittins or any other policy is optimal in the multiserver case. Exactly analyzing the M/G/k under any scheduling policy is intractable, and Gittins is a particularly complicated policy that is hard to analyze even in the single-server case. In this work we introduce monotonic Gittins (M-Gittins), a new variation of the Gittins policy, and show that it minimizes mean response time in the heavy-traffic M/G/k for a wide class of finite-variance job size distributions. We also show that the monotonic shortest expected remaining processing time (M-SERPT) policy, which is simpler than M-Gittins, is a 2-approximation for mean response time in the heavy traffic M/G/k under similar conditions. These results constitute the most general optimality results to date for the M/G/k with unknown job sizes. Our techniques build upon work by Grosof et al. [6], who study simple policies, such as SRPT, in the M/G/k; Bansal et al. [2], Kamphorst and Zwart [7], and Lin et al. [9], who analyze mean response time scaling of simple policies in the heavy-traffic M/G/1; and Aalto et al. [1] and Scully et al. [11, 13], who characterize and analyze the Gittins policy in the M/G/1.

Analytical and simulation studies of queueing-inventory models with MAP demands in batches and positive phase type services

Queueing-inventory models have many practical applications and have been studied extensively in the literature. Most of the studies focus on models in which the demands occur singly. Only a very few papers analyze models wherein the demands occur in batches. In this paper we consider batch demands in the context of two models, both of which assume that the demands occur according to a versatile Markovian point process. The demands need to be serviced with items from the inventory, and the service times are assumed to be of phase type. The replenishment of the inventory is based on the (s, S)-type policy and the lead times are assumed to be random. These two models are such that in the first model an arriving customer finding the inventory level to be zero will be lost; and in the second model a customer can be lost either at the time of an arrival (wherein the server is idle due to zero inventory) or at the time of a service completion (at which time the inventory level becomes zero). In the second model, all waiting customers are removed from the system due to zero inventory. These two models are studied in steady-state using the classical matrix-analytic methods in single server case, and in the case of multi-server systems we resort to simulation using ARENA. Illustrative examples, including an optimization problem, comparing the two models are presented.

Arbitrable blind quantum computation

Blind quantum computation (of a single-server case) is a two-party cryptographic protocol which involves a quantum computation server Bob and a client Alice who wants to delegate her quantum computation to Bob without revealing her quantum algorithms and her input to and output from the algorithms. Since Bob may be truant and pretend to execute some computation, Alice wants to verify Bob’s honesty on computation. To resolve this problem, the notion of the verifiability has been considered in the literature and several protocols of verifiable blind computation have been developed. Verifiable blind quantum computation enables Alice to check whether Bob is cheating or not. In addition to the above problem, another problem could arise. If Alice pretends to be a client and is actually a competitor against Bob, then she might slander Bob by fabricating his dishonesty. Therefore, if either Alice or Bob is cheating, then a “neutral” referee other than Alice and Bob should judge which is honest. The standard definition of the verifiability guarantees that only Alice can verify Bob’s computation, and thus, it should be called private verifiability. If Bob claims his innocence though he is actually cheating, then Alice cannot persuade any others that Bob is really cheating while Alice can recognize Bob’s cheating. In this paper, we incorporate arbitrators as the third party into blind quantum computation to resolve the above problems and give an arbitrable blind quantum computation scheme, which provides public verifiability in some sense.

Advance Reservation Games

Advance reservation (AR) services form a pillar of several branches of the economy, including transportation, lodging, dining, and, more recently, cloud computing. In this work, we use game theory to analyze a slotted AR system in which customers differ in their lead times. For each given time slot, the number of customers requesting service is a random variable following a general probability distribution. Based on statistical information, the customers decide whether or not to make an advance reservation of server resources in future slots for a fee. We prove that only two types of equilibria are possible: either none of the customers makes AR or only customers with lead time greater than some threshold make AR. Our analysis further shows that the fee that maximizes the provider’s profit may lead to other equilibria, one of which yields zero profit. In order to prevent ending up with no profit, the provider can elect to advertise a lower fee yielding a guaranteed but smaller profit. We refer to the ratio of the maximum possible profit to the maximum guaranteed profit as the price of conservatism. When the number of customers is a Poisson random variable, we prove that the price of conservatism is one in the single-server case, but can be arbitrarily high in a many-server system.

Optimal scheduling in call centers with a callback option

We consider a call center model with a callback option, which allows to transform an inbound call into an outbound one. A delayed call, with a long anticipated waiting time, receives the option to be called back. We assume a probabilistic customer reaction to the callback offer (option). The objective of the system manager is to characterize the optimal call scheduling that minimizes the expected waiting and abandonment costs. For the single-server case, we prove that non-idling is optimal. Using a Markov decision process approach, we prove for the two-server case that a threshold policy on the number of queued outbound calls is optimal. For the multi-server case, we numerically characterize a switching curve of the number of agents reserved for inbound calls. It is a function of the number of queued outbound calls, the number of busy agents and the identity of jobs in service. We also develop a Markov chain method to evaluate the system performance measures under the optimal policy.We next conduct a numerical study to examine the impact of the policy parameters on the system performance. We observe that the value of the callback offer is especially important for congested situations. It also appears that the benefits of a reservation policy are more apparent in large call centers, while they almost disappear in the extreme situations of light or heavy workloads. We moreover observe in most cases that the callback offer should be given upon arrival to any delayed call. However, if balking and abandonment are very high (which helps to reduce the workload) or if the overall treatment time spent to serve an outbound call is too large compared to that of an inbound one, there is a value in delaying the proposition of the callback offer.

Improving Queuing System Throughput Using Distributed Mean Value Analysis to Control Network Congestion

In this paper, we have used the distributed mean value analysis (DMVA) technique with the help of random observe property (ROP) and palm probabilities to improve the network queuing system throughput. In such networks, where finding the complete communication path from source to destination, especially when these nodes are not in the same region while sending data between two nodes. So, an algorithm is developed for single and multi-server centers which give more interesting and successful results. The network is designed by a closed queuing network model and we will use mean value analysis to determine the network throughput (b) for its different values. For certain chosen values of parameters involved in this model, we found that the maximum network throughput for β≥0.7 remains consistent in a single server case, while in multi-server case for β≥ 0.5 throughput surpass the Marko chain queuing system.

Performance of popular open source databases for HEP related computing problems

Databases are used in many software components of HEP computing, from monitoring and job scheduling to data storage and processing. It is not always clear at the beginning of a project if a problem can be handled by a single server, or if one needs to plan for a multi-server solution. Before a scalable solution is adopted, it helps to know how well it performs in a single server case to avoid situations when a multi-server solution is adopted mostly due to sub-optimal performance per node. This paper presents comparison benchmarks of popular open source database management systems. As a test application we use a user job monitoring system based on the Glidein workflow management system used in the CMS Collaboration.

Retrial queues with balanced call blending: analysis of single-server and multiserver model

In call centers, call blending consists in the mixing of incoming and outgoing call activity, according to some call blending balance. Recently, Artalejo and Phung-Duc have developed an apt model for such a setting, with a two way communication retrial queue. However, by assuming a classical (proportional) retrial rate for the incoming calls, the short- term blending balance is heavily impacted by the number of incoming calls, which may be undesired, especially when the balance between incoming and outgoing calls is vital to the service offered. In this contribution, we consider an alternative to classical call blending, through a retrial queue with constant retrial rate for incoming calls. For the single-server case (one operator), a generating functions approach enables to derive explicit formulas for the joint stationary distribution of the number of incoming calls and the system state, and also for the factorial moments. This is complemented with a stability analysis, expressions for performance measures, and also recursive formulas, allowing reliable numerical calculation. A correlation study enables to study the system's short-term blending balance, allowing to compare it to that of the system with classical retrial rate. For the multiserver case (multiple operators), we provide a quasi-birth-and-death process formulation, enabling to derive a sufficient and necessary condition for stability in this case (in a simple form), a numerical recipe to obtain the stationary distribution, and a cost model.

AN EFFICIENT METHOD FOR PERFORMANCE ANALYSIS OF BLENDED CALL CENTERS WITH REDIAL

This paper considers a multiserver retrial queue with two-way communication for blended call centers. Primary incoming calls arrive at the servers according to a Poisson process and request an exponentially distributed service time. Incoming calls that find all the servers fully occupied join the orbit and retry to occupy a server again after some random time. A retrial incoming call behaves the same as a primary incoming call. A server not only serves as an incoming call but also makes an outgoing call after some random idle time. We assume that the distribution of the duration of outgoing calls is different from that of incoming calls. Artalejo and Phung-Duc (2012) have extensively studied the single server case and have obtained some preliminary results for the multiserver model. In this paper, we present an extensive analysis for the multiserver model in which, we propose a new formulation by a level-dependent quasi birth-and-death (QBD) process, whose block matrices have some special block structure. Based on a matrix continued fraction approach and a censoring technique, we develop an efficient algorithm utilizing the special structure for computing the stationary distribution. Furthermore, we derive explicit formulae for the mean number of incoming calls and that of outgoing calls in the servers. Through various numerical results, we study the characteristics of the queueing system and find some insights into the optimal outgoing call rate.

A mechanism for pricing and resource allocation in peer-to-peer networks

In this study, we design a pricing and allocation mechanism for a peer-to-peer (P2P) network that allows users in a firm to effectively share computing resources. This mechanism allows tasks (incoming jobs) to be allocated to resources (participating peers) in an organization in a decentralized manner. The base case of our P2P mechanism derives the optimal transfer price that maximizes the net value, which is characterized as the difference between the expected gross value of the jobs and the expected delay cost, for the peers in the network. The optimal price for executing any job at a peer location is essentially equal to the marginal delay cost it imposes on all current jobs at that node. With this pricing scheme, no individual user has an incentive to over-utilize shared resources, thereby avoiding the “tragedy of the commons” for the P2P network. Our model builds on the classical Mendelson (1985) study that was one of the first to look at the control and management issues related to a single-server computer system. In this study, we model transfer pricing for a multiple server environment of a P2P network. The original Mendelson (1985) model thus becomes a special single-server case of our general model. Our basic model is extended to incorporate situations where the peers have queue length constraints, which may be used for providing quality of service (QoS) guarantees to users. We then perform numerical computations that illustrate the effects of job arrivals on prices and job allocations at the individual servers. There is an enormous potential for P2P based technologies to help organizations manage their computer resources effectively.

Location of single-server immobile facilities subject to a lossconstraint

Waiting may be unacceptable, even a short time, at a facility providing a serviceinvolving medical or other emergencies. Hence, it is appropriate to locate suchfacilities so that the rate at which users are lost is limited. Eachfacility will here be modelled as anM/Er/m/N queueing systemsubject to a loss restriction constraint and the single-server case(m=1) will be treated in detail. Introduction of the Erlangdistribution for service times allows a better fit of the model to actual valuesof both mean and variance than do currently available models that use anexponential distribution. Location of facilities will be such that the averagetravel time to a facility is minimized. It is shown how a deterministicconstraint, equivalent to the loss constraint, can be generated resulting in aninteger linear program, and values of a parameterρc which facilitates this linearization aretabulated for various values of r, N and service leveldemanded. Numerical experiments are performed including an application looselyrelated to the location of neonatal clinics in the Municipality of Rio deJaneiro. Finally, there is a discussion of how further improved modelling of theservice time distribution might be effected.

Algorithmic Analysis of the Maximum Queue Length in a Busy Period for the M/M/c Retrial Queue

This paper deals with the maximum number of customers in orbit (and in the system) during a busy period for the M/M/c retrial queue. Determining the distribution for the maximum number of customers in orbit is reduced to computation of certain absorption probabilities. By reducing to the single-server case we arrive at a closed analytic formula. For the multi-server case we develop an efficient algorithmic procedure for computation of this distribution by exploiting the special block-tridiagonal structure of the system. Numerical results illustrate the efficiency of the method and reveal interesting facts concerning the behavior of the M/M/c retrial queue.

Parallel Scheduling of Multiclass M/M/m Queues: Approximate and Heavy-Traffic Optimization of Achievable Performance

We address the problem of scheduling a multiclass M/M/m queue with Bernoulli feedback on m parallel servers to minimize time-average linear holding costs. We analyze the performance of a heuristic priority-index rule, which extends Klimov's optimal solution to the single-server case: servers select preemptively customers with larger Klimov indices. We present closed-form suboptimality bounds (approximate optimality) for Klimov's rule, which imply that its suboptimality gap is uniformly bounded above with respect to (i) external arrival rates, as long as they stay within system capacity; and (ii) the number of servers. It follows that its relative suboptimality gap vanishes in a heavy-traffic limit, as external arrival rates approach system capacity (heavy-traffic optimality). We obtain simpler expressions for the special no-feedback case, where the heuristic reduces to the classical cμ rule. Our analysis is based on comparing the expected cost of Klimov's rule to the value of a strong linear programming (LP) relaxation of the system's region of achievable performance of mean queue lengths. In order to obtain this relaxation, we derive and exploit a new set of work decomposition laws for the parallel-server system. We further report on the results of a computational study on the quality of the cμ rule for parallel scheduling.

Analysis of Markov Multiserver Retrial Queues with Negative Arrivals

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. In the paper we analyze multiserver retrial queues; i.e., any customer finding all servers busy upon arrival must leave the service area and re-apply for service after some random time. The control mechanism is such that, whenever the service facility is full occupied, an exponential timer is activated. If the timer expires and the service facility remains full, then a random batch of customers, which are stored at the retrial pool, are automatically removed. This model extends the existing literature, which only deals with a single server case and individual removals. Two different approaches are considered. For the stable case, the matrix–analytic formalism is used to study the joint distribution of the service facility and the retrial pool. The approximation by more simple infinite retrial model is also proved. In the overloading case we study the transient behaviour of the trajectory of the suitably normalized retrial queue and the long-run behaviour of the number of busy servers. The method of investigation in this case is based on the averaging principle for switching processes.

The cache location problem

This paper studies the problem of where to place network caches. Emphasis is given to caches that are transparent to the clients since they are easier to manage and they require no cooperation from the clients. Our goal is to minimize the overall flow or the average delay by placing a given number of caches in the network. We formulate these location problems both for general caches and for transparent en-route caches (TERCs), and identify that, in general, they are intractable. We give optimal algorithms for line and ring networks, and present closed form formulae for some special cases. We also present a computationally efficient dynamic programming algorithm for the single server case. This last case is of particular practical interest. It models a network that wishes to minimize the average access delay for a single web server. We experimentally study the effects of our algorithm using real web server data. We observe that a small number of TERCs are sufficient to reduce the network traffic significantly. Furthermore, there is a surprising consistency over time in the relative amount of web traffic from the server along a path, lending a stability to our TERC location solution. Our techniques can be used by network providers to reduce traffic load in their network.

Parallel Scheduling of Multiclass M/M/m Queues: Approximate and Heavy-Traffic Optimization of Achievable Performance

We address the problem of scheduling a multiclass M/M/m queue with Bernoulli feedback on m parallel servers to minimize time-average linear holding costs. We analyze the performance of a heuristic priority-index rule, which extends Klimov's optimal solution to the single-server case: servers select preemptively customers with larger Klimov indices. We present closed-form suboptimality bounds (approximate optimality) for Klimov's rule, which imply that its suboptimality gap is uniformly bounded above with respect to (i) external arrival rates, as long as they stay within system capacity; and (ii) the number of servers. It follows that its relative suboptimality gap vanishes in a heavy-traffic limit, as external arrival rates approach system capacity (heavy-traffic optimality). We obtain simpler expressions for the special no-feedback case, where the heuristic reduces to the classical c mu rule. Our analysis is based on comparing the expected cost of Klimov's rule to the value of a strong linear programming (LP) relaxation of the system's region of achievable performance of mean queue lengths. In order to obtain this relaxation, we derive and exploit a new set of work decomposition laws for the parallel-server system. We further report on the results of a computational study on the quality of the c mu rule for parallel scheduling.

PERFORMANCE OF A HYSTERESIS BASED CONTROL STRATEGY FOR A FLEXIBLE MACHINE OPERATING UNDER A PERIODIC STATUS MONITORING POLICY

The role of on-line control strategies is crucial for exploiting the flexibility within discrete part manufacturing systems. When studying their impact on system performance, researchers have mostly assumed the presence of real-time information availability. Such assumption may not always be valid in real world manufacturing situations since on-line control strategies base their decisions on information that may entail significant time delays depending upon the extent of computer based automation and information integration employed. We refer to these delays as information delays. In the domain of discrete part manufacturing systems with defined levels of flexibility, it is essential explicitly to model and analyze the effect that information delays have on the performance of a given on-line control strategy. In this paper we conceptualize three basic modes of information delays and indicate the need to model them explicitly in order to assess their impact on the performance of on-line control strategies. One of these basic forms of information delay is caused due to a review period based status monitoring policy. We propose and characterize a hysteresis based on-line control strategy (HCS) to cope with information delays. The study system chosen represents the simplest possible manifestation of machine flexibility and resembles the two-queue single server case for which prior research results indicate the superiority of the alternating priority (AP) rule when average job flowtime is the measure of performance. We use computer simulation to compare the behaviour of HCS with the AP rule in a real-time as well as the information delay mode. The best HCS setting in the real-time mode fails to remain consistent in the information delay case, thus clearly indicating the need to explicitly model delays when developing suitable control strategies. Further, when compared with the AP rule, HCS shows consistently superior performance to AP. © 1997 Elsevier Science Ltd