Average Sojourn Time Research Articles

On queueing systems with a fractional number of devices

Queueing systems with several servers and a special discipline whereby demands are successively received by servers until the servicing is completed are considered. Among these systems, there are such systems whose characteristics constantly depend on a certain positive parameter. Given integer values of the latter, the waiting probability and average sojourn time coincide with those calculated using the Erlang C formula. Therefore, these systems can be considered as generalizations of the classical M/M/s system to a fractional number of servers.

Performance analysis of a discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper presents the analysis of a discrete-time Geo/G/1 queue with randomized vacations. Using the probability decomposition theory and renewal process, two variants on this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both the cases, recursive solution for queue length distributions at arbitrary, just before a potential arrival, pre-arrival, immediately after potential departure, and outside observer’s observation epochs are obtained. Further, various performance measures such as potential blocking probability, turned-on period, turned-off period, vacation period, expected length of the turned-on circle period, average queue length and sojourn time, etc. have been presented. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.

The mechanics of stochastic slowdown in evolutionary games

We study the stochastic dynamics of evolutionary games, and focus on the so-called ‘stochastic slowdown’ effect, previously observed in Altrock et al. (2010) for simple evolutionary dynamics. Slowdown here refers to the fact that a beneficial mutation may take longer to fixate than a neutral one. More precisely, the fixation time conditioned on the mutant taking over can show a maximum at intermediate selection strength. We show that this phenomenon is present in the Prisoner's Dilemma, and also discuss counterintuitive slowdown and speedup in coexistence games. In order to establish the microscopic origins of these phenomena, we calculate the average sojourn times. This allows us to identify the transient states which contribute most to the slowdown effect, and enables us to provide an understanding of slowdown in the takeover of a small group of cooperators by defectors in the Prisoner's Dilemma: Defection spreads fast initially, but the final steps to takeover can be delayed substantially. The analysis of coexistence games reveals even more intricate non-monotonic behavior. In small populations, the conditional average fixation time can show multiple extrema as a function of the selection strength, e.g., slowdown, speedup, and slowdown again. We classify generic 2×2 games with respect to the possibility to observe non-monotonic behavior of the conditional average fixation time as a function of selection strength.

Load balancing via random local search in closed and open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach themselves to an arbitrary server, but may switch servers independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner. We first analyze the natural Random Local Search (RLS) strategy. Under this strategy, after sampling a new server randomly, clients only switch to it if their service rate is improved. In closed systems, where the client population is fixed, we derive tight estimates of the time it takes under RLS strategy to balance the load across servers. We then study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures. We compare the load-aware RLS strategy to a load-oblivious strategy in which clients just randomly switch server without accounting for the server loads. Surprisingly, we show that both load-oblivious and load-aware strategies stabilize the system whenever this is at all possible. We use large-system asymptotics to characterize system performance, and augment this with simulations, which suggest that the average client sojourn time under the load-oblivious strategy is not considerably reduced when clients apply smarter load-aware strategies.

Qualitative study of a quarantine/isolation model with multiple disease stages

Recent studies suggest that, for disease transmission models with latent and infectious periods, the use of gamma distribution assumption seems to provide a better fit for the associated epidemiological data in comparison to the use of exponential distribution assumption. The objective of this study is to carry out a rigorous mathematical analysis of a communicable disease transmission model with quarantine (of latent cases) and isolation (of symptomatic cases), in which the waiting periods in the infected classes are assumed to have gamma distributions. Rigorous analysis of the model reveals that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique endemic equilibrium when the threshold quantity exceeds unity. The endemic equilibrium is shown to be locally and globally-asymptotically stable for special cases. Numerical simulations, using data related to the 2003 SARS outbreaks, show that the cumulative number of disease-related mortality increases with increasing number of disease stages. Furthermore, the cumulative number of new cases is higher if the asymptomatic period is distributed such that most of the period is spent in the early stages of the asymptomatic compartments in comparison to the cases where the average time period is equally distributed among the associated stages or if most of the time period is spent in the later (final) stages of the asymptomatic compartments. Finally, it is shown that distributing the average sojourn time in the infectious (asymptomatic) classes equally or unequally does not effect the cumulative number of new cases.

Analysis of multiclass Markovian polling systems with feedback and composite scheduling algorithms

We consider multiclass Markovian polling systems with feedback and analyze their average performance measures. Scheduling in polling systems has many applications in computer and communication systems. We utilize the framework that has been effectively used to analyze various composite scheduling algorithms in many types of multiclass queues systematically in conjunction with the functional computation method (Hirayama in Naval Research Logistics 50:719–741, 2003; Journal of the Operations Research Society of Japan 48:226–255, 2005; Advances in queueing theory and network applications, pp. 119–146, Springer, New York, 2009a; Journal of Industrial and Management Optimization 6:541–568, 2010). We define the conditional expected values of the performance measures such as the sojourn times as functions of the system state and find their expressions by solving some equations. Then from these expressions, we derive the average numbers of customers and the average sojourn times for all service stages of customers circulating the system. We consider their application to a packet scheduling problem where multiple categories of packets share a resource.

On the stability and optimality of universal swarms

Recent work on BitTorrent swarms has demonstrated that a bandwidth bottleneck at the seed can lead to the underutilization of the aggregate swarm capacity. Bandwidth underutilization also occurs naturally in mobile peer-to-peer swarms, as a mobile peer may not always be within the range of peers storing the content it desires. We argue in this paper that, in both cases, idle bandwidth can be exploited to allow content sharing across multiple swarms, thereby forming a universal swarm system. We propose a model for universal swarms that applies to a variety of peer-to-peer environments, both mobile and online. Through a fluid limit analysis, we demonstrate that universal swarms have significantly improved stability properties compared to individually autonomous swarms. In addition, by studying a swarm's stationary behavior, we identify content replication ratios across different swarms that minimize the average sojourn time in the system. We then propose a content exchange scheme between peers that leads to these optimal replication ratios, and study its convergence numerically.

Distributive Network Utility Maximization Over Time-Varying Fading Channels

Distributed network utility maximization (NUM) has received an increasing intensity of interest over the past few years. Distributed solutions (e.g., the primal-dual gradient method) have been intensively investigated under fading channels. As such distributed solutions involve iterative updating and explicit message passing, it is unrealistic to assume that the wireless channel remains unchanged during the iterations. Unfortunately, the behavior of those distributed solutions under time-varying channels is in general unknown. In this paper, we shall investigate the convergence behavior and tracking errors of the iterative primal-dual scaled gradient algorithm (PDSGA) with dynamic scaling matrices (DSC) for solving distributive NUM problems under time-varying fading channels. We shall also study a specific application example, namely the multi-commodity flow control and multi-carrier power allocation problem in multi-hop ad hoc networks. Our analysis shows that the PDSGA converges to a limit region rather than a single point under the finite state Markov chain (FSMC) fading channels. We also show that the order of growth of the tracking errors is given by O(T/N), where T and N are the update interval and the average sojourn time of the FSMC, respectively. Based on this analysis, we derive a low complexity distributive adaptation algorithm for determining the adaptive scaling matrices, which can be implemented distributively at each transmitter. The numerical results show the superior performance of the proposed dynamic scaling matrix algorithm over several baseline schemes, such as the regular primal-dual gradient algorithm.

Dynamic Channel Adaptation for IP Based Split Spectrum Femto/Macro Cellular Systems

In femto/macro cellular networks, the stability and fairness problems caused by the unplanned and random characteristic of femtocells must be solved. By applying queueing theory in IP based femto/macro cellular networks, we found the stability condition, and described two kinds of cell section policies of users. As a main contribution, we provided the adaptive channel distribution algorithm which minimizes the average packet sojourn time at transmitting systems and keeps the whole systems stable and fair among cells. Through experiments in various environments, we analyzed the influence of channel reuse factor, cell selection policies, and the number of femtocells on system performance.

RATIO BETWEEN AVERAGE SOJOURN TIMES UNDER PROCESSOR SHARING AND FAIR SOJOURN PROTOCOL

Under very general assumptions, we prove that the ratio between the average sojourn time in a queue under the processor sharing (PS) and the same queue under the fair sojourn protocol (FSP) can be bounded above, and we derive such a bound in terms of the long-run average system size of the PS queue seen by arrivals.

Load balancing via random local search in closed and open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach to an arbitrary server, but may switch servers independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner. We first analyze the natural Random Local Search (RLS) strategy. Under this strategy, after sampling a new server randomly, clients only switch to it if their service rate is improved. In closed systems, where the client population is fixed, we derive tight estimates of the time it takes under RLS strategy to balance the load across servers. We then study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures. We compare the load-aware RLS strategy to a load-oblivious strategy in which clients just randomly switch server without accounting for the server loads. Surprisingly, we show that both load-oblivious and load-aware strategies stabilize the system whenever this is at all possible. We further demonstrate, using large-system asymptotics, that the average client sojourn time under the load-oblivious strategy is not considerably reduced when clients apply smarter load-aware strategies.

Distributive Power Control Algorithm for Multicarrier Interference Network Over Time-Varying Fading Channels—Tracking Performance Analysis and Optimization

Distributed power control over interference limited network has received an increasing intensity of interest over the past few years. Distributed solutions (like the iterative water-filling, gradient projection, etc.) have been intensively investigated under \emph{quasi-static} channels. However, as such distributed solutions involve iterative updating and explicit message passing, it is unrealistic to assume that the wireless channel remains unchanged during the iterations. Unfortunately, the behavior of those distributed solutions under \emph{time-varying} channels is in general unknown. In this paper, we shall investigate the distributed scaled gradient projection algorithm (DSGPA) in a $K$ pairs multicarrier interference network under a finite-state Markov channel (FSMC) model. We shall analyze the \emph{convergence property} as well as \emph{tracking performance} of the proposed DSGPA. Our analysis shows that the proposed DSGPA converges to a limit region rather than a single point under the FSMC model. We also show that the order of growth of the tracking errors is given by $\mathcal{O}\(1 \big/ \bar{N}\)$, where $\bar{N}$ is the \emph{average sojourn time} of the FSMC. Based on the analysis, we shall derive the \emph{tracking error optimal scaling matrices} via Markov decision process modeling. We shall show that the tracking error optimal scaling matrices can be implemented distributively at each transmitter. The numerical results show the superior performance of the proposed DSGPA over three baseline schemes, such as the gradient projection algorithm with a constant stepsize.

The M=M=1 Queue with Single Working Vacation Serving at a Slower Rate during the Start-up Period

We consider the M/M/1 queue with single working vacation serving at a slower rate during the start-up period. In orderto save energy and reduce waste, the server works at a slower rate rather than completely stops during a vacation andstart-up period. Using quasi birth and death process and matrix-geometric solution method, we obtain the distribution ofthe nember of customers in the system, the average nember of the customers and the average sojourn time of a customerin the stationary state.

Autonomic Clouds on the Grid

Computational clouds constructed on top of existing Grid infrastructure have the capability to provide different entities with customized execution environments and private scheduling overlays. By designing these clouds to be autonomically self-provisioned and adaptable to changing user demands, user-transparent resource flexibility can be achieved without substantially affecting average job sojourn time. In addition, the overlay environment and physical Grid sites represent disjoint administrative and policy domains, permitting cloud systems to be deployed non-disruptively on an existing production Grid. Private overlay clouds administered by, and dedicated to the exclusive use of, individual Virtual Organizations are termed Virtual Organization Clusters. A prototype autonomic cloud adaptation mechanism for Virtual Organization Clusters demonstrates the feasibility of overlay scheduling in dynamically changing environments. Commodity Grid resources are autonomically leased in response to changing private scheduler loads, resulting in the creation of virtual private compute nodes. These nodes join a decentralized private overlay network system called IPOP (IP Over P2P), enabling the scheduling and execution of end user jobs in the private environment. Negligible overhead results from the addition of the overlay, although the use of virtualization technologies at the compute nodes adds modest service time overhead (under 10%) to computationally-bound Grid jobs. By leasing additional Grid resources, a substantial decrease (over 90%) in average job queuing time occurs, offsetting the service time overhead.

Service optimization with patient satisfaction in healthcare systems

Market research on services and waiting lends support to the fact that waiting for service is an undesirable phenomenon adversely affecting customer satisfaction and consequently business enterprises. The simulation optimization strategy described in this study addresses both the subjective as well as the objective elements in the patients’ evaluation of healthcare services. The cost function simulates the average sojourn time of the patients in the healthcare system, whereas the fuzzy constraints quantitatively estimate the overall waiting and service experience of the patients. Minimization of the cost function (objective element in the patients’ evaluation of service) subject to the fuzzy constraints (subjective elements in the patients’ evaluation of service) leads to an increased patient satisfaction, increased efficiency of system operation and consequently increased profits. The proposed method makes a topological cum functional model of the healthcare system, simulates the operation via discrete event simulation and optimizes it using the Genetic Algorithm.

Modeling and performance evaluation of the sleep mode in the IEEE 802.16e wireless networks with self-similar traffic

IEEE 802.16e is currently the latest broadband wireless access standard designed to support mobility. In mobile networks, how to control energy consumption is one of the most important issues for battery-powered mobile stations. The standard proposes an energy saving mechanism named ‘sleep mode’ for conserving the power of mobile stations. According to the operation principle of the sleep mode for downlink traffic in the type I power saving class, considering the self-similar nature of massive multimedia data in wireless networks, a discrete-time batch arrival Geom X /G/1 queuing model with a close-down time and multiple vacations is built. The batch size is supposed to be Pareto distributed. By employing an embedded Markov chain method, the average queue length and the average sojourn time of the system model are derived. Correspondingly, the performance measures are obtained of the energy saving rate and the average packet delay time for the sleep mode in the IEEE 802.16e. The numerical results demonstrate the dependency relationships between the system performance measures and the system parameters with different offered loads and different self-similar degrees. Furthermore, a cost model is developed to determine the optimum length of the close-down time for minimizing the total system cost.

Modelling and simulation of B3G multi-service traffic in the presence of mobility

A multi-service traffic model is proposed, and its validation is achieved by using event-based simulation results which consider the burstiness of traffic. By considering the deployment scenarios and tele-traffic parameters from the Vehicular scenario of IST-SEACORN, a simulator was produced to extract conclusions about blocking and handover failure probabilities. Simulations, were performed for different cases, from single- to multi-service situations, and from absence to presence of mobility. Besides results for the quality of service in the air interface, including blocking and handover failure probabilities for several types of traffic, the simulator allows for extracting conclusions about the validation of the Bernoulli/Poisson/Pascal model for the computation of the on/off blocking probability, the ratio between the number of session rejected at the beginning of an on period and the total number of bursts generated during a session. In the single-service case, the theoretical and the experimental values of on/off blocking probability are close to each other. An almost perfect concordance between theoretical and simulation values is verified when the average sojourn time in cells is equal to the average holding time. In the multi-service case, although the results are not exactly coincident a coherent behaviour is achieved for an average traffic per user up to 0.06 Erl.

Performance evaluation for the sleep mode in the IEEE 802.16e based on a queueing model with close-down time and multiple vacations

IEEE 802.16e is the latest broadband wireless access standard designed to support mobility. In mobile networks, how to control energy consumption is one of the most important issues for the battery-powered mobile stations. The standard proposes an energy saving mechanism that named for conserving the power of the mobile stations. According to the operation mechanism of the sleep mode for downlink traffic in the type I power saving class, a discrete-time Geom/G/1 queueing model with close-down time and multiple vacations is built. By employing an embedded Markov chain method and Little's law, the average queue length, the average sojourn time and the average busy cycle of the queueing model are derived. Correspondingly, we get the performance measures of the energy saving rate and the average packet delay time for the sleep mode in the IEEE 802.16e. Finally, numerical results are given to demonstrate the dependency relationships between the system performance measures and the system parameters. Furthermore, a cost model is developed to determine the optimum length of the close-down time for minimizing the total system cost.

The Endogenous Analysis of Flows, with Applications to Migrations, Social Mobility and Opinion Shifts

Two major traditions coexist in the statistical analysis of spatial and social change: mobility analysis (e.g., Boudon, 1973) initiated by Prais (1953) on one hand, and gravity modelling for geographic flows (e.g., Fotheringham and O'Kelly, 1989; Sen and Smith, 1995), initiated by Reilly (1931) on the other hand. This paper focuses on the formal content of both traditions and explores their relationship. Three small data sets (migrations, occupational mobility and opinion shifts) illustrate the issues. In particular, we show how the sojourn times and jump matrices of mobility analysis are related to the accessibilities and expansivities of gravity modelling. We also address in detail the issues of time-dependence and categories aggregation. Two well-behaved indices of mobility are defined and discussed, namely the average sojourn time, which measures short-time mobility, and the second eigenvalue of the transition matrix measuring long-time mobility.

Modeling, scheduling, and simulation of switched processing systems

Switched Processing Systems (SPS) serve as canonical models in a wide area of applications such as high performance computing, wireless networking, call centers, and flexible manufacturing. In this article, we model the SPS by considering both slotted and continuous time and analyze it under fairly mild stochastic assumptions. Two classes of scheduling policies are introduced and shown to maximize the throughput and maintain strong stability of the system. In addition, their performance with respect to the average job sojourn time is examined by simulating small SPS subject to different types of input traffic. By utilizing the simulation result of the proposed policies, a hybrid control policy is constructed to reduce the average job sojourn time when the system has unknown and changing input loads.