Exponential Service Times Research Articles

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

We investigate a two-station queueing system where strategic customers must be sequentially serviced at both stations. We prove that an established property — that the distribution of the total time spent in a two-station system is independent of the chosen route when services times are exponentially distributed — is not a general one by providing a counterexample with deterministic service times. In doing so, we also prove a concomitant property — that any routing strategy is an equilibrium — is peculiar to a system with a particular assumption of an exponential service time distribution. Using simulations, we show that — depending on the distribution of service times — there can be (1) cases with three equilibria, (2) cases with one pure strategy equilibrium, and (3) cases with one mixed strategy equilibrium.

Dynamic load balancing in energy packet networks

Energy Packet Networks (EPNs) model the interaction between renewable sources generating energy following a random process and communication devices that consume energy. This network is formed by cells and, in each cell, there is a queue that handles energy packets and another queue that handles data packets. We assume Poisson arrivals of energy packets and of data packets to all the cells and exponential service times. We consider an EPN model with a dynamic load balancing where a cell without data packets can poll other cells to migrate jobs. This migration can only take place when there is enough energy in both interacting cells, in which case a batch of data packets is transferred and the required energy is consumed (i.e. it disappears). We consider that data packet also consume energy to be routed to the next station. Our main result shows that the steady-state distribution of jobs in the queues admits a product form solution provided that a stable solution of a fixed point equation exists. We prove sufficient conditions for irreducibility. Under these conditions and when the fixed point equation has a solution, the Markov chain is ergodic. We also provide sufficient conditions for the existence of a solution of the fixed point equation. We then focus on layered networks and we study the polling rates that must be set to achieve a fair load balancing, i.e., such that, in the same layer, the load of the queues handling data packets is the same. Our numerical experiments illustrate that dynamic load balancing satisfies several interesting properties such as performance improvement or fair load balancing.

Poisson Input and Exponential Service Time Finite Capacity Interdependent Queueing Model with Breakdown and Controllable Arrival Rates

Objectives: This study aims at (i) introducing the finite capacity of the interdependent queueing model with breakdown and controllable arrival rates, (ii) calculating the average number of clients in the system, and identifying the expected waiting period of the clients in the system, (iii) dealing with the model descriptions, steady-state equations, and characteristics, which are expressed in terms of , and (iv) analyzing the probabilities of the queueing system and its characteristics with numerical verification of the obtained results. Methods: While providing the input, the arrival rates through faster and slower arrival rates are controlled using the Poisson process. Also, the service provides an exponential distribution. The server provides the service on an FCFS basis. In this article, two types of models are used: and which are the system’s conditions, where represents the number of units present in the queue in which their probability is and . All probabilities are distributed based on the speed of advent using this concept. Then, the steady-state probabilities are computed using a recursive approach. Findings: This paper discovers the number of clients using the system on average and the expected number of clients in the system using the probability of the steady-state calculation. Little’s formula is used to derive the expected waiting period of the clients in the system. Novelty: There are articles connected to the finite capacity of failed service in functioning and malfunctioning, but this takes the initiative to provide a link in connection with the rates of the controllable arrivals and interdependency in the arrival and service processes. Mathematics Subject allocation: 60K25, 68M20, 90B22. Keywords: M/M/1/K Queue Model, Finite Capacity, Breakdown, Controllable Arrival rates, FCFS Queue Discipline

Whittle index approach to multiserver scheduling with impatient customers and DHR service times

We consider the optimal scheduling problem in a multiserver queue with impatient customers belonging to multiple classes. We assume that each customer has a random abandonment time, after which the customer leaves the system if its service has not been completed before that. In addition, we assume that the scheduler is not able to anticipate the expiration of the abandonment times but only knows their distributions and how long each customer has been in the system. Many papers consider this scheduling problem under Poisson arrivals and linear holding costs assuming further that both the service times and the abandonment times have exponential distributions. Even with these additional assumptions, the exact solution is known only in very few special cases. To tackle this tricky problem, we apply the Whittle index approach. Unlike the earlier papers, which were restricted to exponential service times, we allow the service time distributions for which the hazard rate is decreasing. The Whittle index approach is applied to the discrete-time multiserver queueing problem with discounted costs. As our main theoretical result, we prove that the related relaxed optimization problem is indexable and derive the corresponding Whittle index explicitly. Based on this discrete-time result, we develop a reasonable heuristic for the original continuous-time multiserver scheduling problem. The performance of the resulting policy is evaluated in the M/G/M setup by numerical simulations, which demonstrate that it, indeed, gives better performance than the other policies included in the comparison.

Optimal sequencing using a scheduling heuristic

The problem of finding the optimal appointment schedule has received significant consideration in the literature on healthcare optimization. Due to randomness, determining when clients should be scheduled in order to strike a balance in terms of idle times, waiting times, and overtime is a non-trivial task. Furthermore, it is often solely studied in the dimension of scheduling, whereas the sequencing decision in which heterogeneous clients (e.g., new and return clients) are to arrive is at least of equal importance. The approach presented integrates an approximation method with a scheduling heuristic that tackles this joint objective of sequencing and scheduling.A moment-iteration method is employed to construct appointment schedules. This method permits analytical solutions and thereby is incredibly fast to solve and optimize. Next, a sequential optimization approach is applied which aligns well with the moment-iteration method and permits per-client control of costs. Moreover, within this approach, the classical surgery scheduling problem of minimizing earliness and tardiness by determining due dates is equivalent to the appointment scheduling problem, i.e., in a sequentially optimal schedule the inter-arrival time equals the previous client’s due date.Finally, for the case of clients having heterogeneous service-time distributions, rigorous sequencing rules are derived in terms of the first two moments. The focus lies on non-identical exponential service times and log-normal service times, as commonly found in healthcare modelling. The smallest-mean/variance-first rule is sequentially optimal. For cases in which the mean and variance move in opposite directions contour plots are provided to aid practitioners in sequencing clients.

Математическая модель расчета и анализа характеристик оптического коммутатора в переходном режиме

The article considers a transient behavior of an optical switch described using a single-channel finite buffer queuing system with Poisson input stream and exponential service time distribution. To solve the Kolmogorov equations system and find the probabilities of states in the transition mode, the apparatus of the Laplace transform was used. Analytical expressions are found for determining the probability of losses and the throughput of an optical switch at a given point in time. In a numerical example, the performance metrics of an optical switch with a buffer size of two packets are analyzed for different network loads in transient and stationary modes. The stationary performance metrics obtained using the proposed approach coincided with the known results, which confirms the correctness of the conclusions. The optical switch transient behavior analyses at different network load showed that with a decreasing service rate the transition mode time increases.

Instability of LRTF multiclass queueing networks

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the longest remaining time first (LRTF) service discipline. It is a reentrant line with two servers and four customer classes, Poisson arrivals and deterministic service times. Our instability proof can be generalized to a class of light-tailed stochastic model primitives, including independent, identically distributed exponential service times. A similar example of an unstable deterministic, strictly subcritical LRTF network is also given. Our arguments work for any service discipline such that there are no departures from any network station during its busy period.

Vessels Arrival Process and its Application to the SHIP/M/infty Queue

In modeling of port dynamics it seems reasonable to assume that the ships arrive on a somewhat scheduled basis and that there is a constant lay period during which, in a uniform way, each vessel can arrive at the port. In the present paper, we study the counting process N(t) which represents the number of scheduled vessels arriving during the time interval (0, t], t>0. Specifically, we provide the explicit expressions of the probability generating function, the probability distribution and the expected value of N(t). In some cases of interest, we also obtain the probability law of the stationary counting process representing the number of arrivals in a time interval of length t when the initial time is an arbitrarily chosen instant. This leads to various results concerning the autocorrelations of the random variables X_i, iin mathbb {Z}, which give the actual interarrival time between the (i-1)-th and the i-th vessel arrival. Finally, we provide an application to a stochastic model for the queueing behavior at the port, given by a queueing system characterized by stationary interarrival times X_i, exponential service times and an infinite number of servers. In this case, some results on the average number of customers and on the probability of an empty queue are disclosed.

Efficient Distributed Threshold-Based Offloading for Large-Scale Mobile Cloud Computing

Mobile cloud computing enables compute-limited mobile devices to perform real-time intensive computations such as speech recognition or object detection by leveraging powerful cloud servers. An important problem in large-scale mobile cloud computing is computational offloading, where each mobile device decides when and how much computation should be uploaded to cloud servers by considering the local processing delay and the cost of using cloud servers. In this paper, we develop a distributed threshold-based offloading algorithm where it uploads an incoming computing task to cloud servers if the number of tasks queued at the device reaches the threshold and processes it locally otherwise. The threshold is updated iteratively based on the computational load and the cost of using cloud servers. We formulate the problem as a symmetric game, and characterize the sufficient and necessary conditions for the existence and uniqueness of the Nash Equilibrium (NE) assuming exponential service times. Then, we show the convergence of our proposed distributed algorithm to the NE when the NE exists. Further, we characterize the performance gap between cost under our proposed distributed algorithm and the minimum cost in terms of Price of Anarchy (PoA) when the cost of using cloud servers is high. Finally, we perform extensive simulations to validate our theoretical findings, demonstrate the efficiency of our proposed distributed algorithm under various scenarios such as hyperexponential service times, imperfect server utilization estimation, and asynchronous threshold updates, and reveal the superior performance of threshold-based policies over their probabilistic counterpart.

Admission Control Policies in Loss Networks

This article addresses the admission control of a loss queueing network, or shortly loss network, of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> parallel multiserver stations with no waiting rooms. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> classes of customers arrive at a Poisson rate and require an exponential service time with common mean. A customer can be served by any free server of a station in the set of stations determined by the class of the customer. Admission of a customer to a station brings different rewards that depend on customer class and the preference order of that station. Our objective is to find the optimal admission control policy that maximizes the average reward. We adopt a research strategy that evolves from simpler networks to more complicated ones. First, we consider a one-station <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -class loss network whose results form a fundamental basis for the analysis of more complicated networks. Second, we consider a two-station <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -class loss network for which we establish the existence of an optimal threshold admission policy. Finally, we consider a general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> -station <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -class loss network and propose an iterative approximation policy (IAP) and a mixed-integer linear programming model that is proven to compute an upper bound. We demonstrate the near-optimal performance of the proposed IAP and the tightness of the upper bound on several numerical instances motivated by real-life networks, such as healthcare emergency service networks and emergency call centers. Note to Practitioners—This article is mainly motivated by real-time scheduling of emergency service networks, such as emergency healthcare and telecommunication networks. In such networks, emergent demands need to be served promptly and are lost otherwise. They can be served by multiple hospitals/operator groups but have their own preferences. The decision maker has to decide dynamically whether to keep the capacity of a hospital/operator for its own customers to meet customer preference or serve the incoming demand to improve the resource utilization. We set the problem as the admission control of a multiclass multistation loss network, study the properties of the optimal policy, and propose an iterative approximation policy that is proven to be near-optimal for large-size loss networks. The proposed policy is found to improve by up to 16% and 52% the no-control policy admitting the customer to the most preferred available station and the no-overflow policy with admission to only the most preferred station, respectively. Managerial insights and extensions are also discussed.

Optimization of Open Queuing Networks with Batch Services

In this paper, open queuing networks with Poisson arrivals and single-server infinite buffer queues are considered. Unlike traditional queuing models, customers are served (with exponential service time) in batches, so that the nodes are non-work-conserving. The main contribution of this work is the design of an efficient algorithm to find the batch sizes which minimize the average response time of the network. As preliminary steps at the basis of the proposed algorithm, an analytical expression of the average sojourn time in each node is derived, and it is shown that this function, depending on the batch size, has a single minimum. The goodness of the proposed algorithm and analytical formula were verified through a discrete-event simulation for an open network with a non-tree structure.

Modified Erlang Loss System for Cognitive Wireless Networks

This paper considers a modified Erlang loss system for cognitive wireless networks and related applications. A primary user has pre-emptive priority over secondary users, and the primary customer is lost if upon arrival all the channels are used by other primary users. Secondary users cognitively use idle channels, and they can stay (either in an infinite buffer or in an orbit) in cases where idle channels are not available upon arrival or they are interrupted by primary users. While the infinite buffer model represents the case with zero sensing time, the infinite orbit model represents the case with positive sensing time. We obtain an explicit stability condition for the cases where arrival processes of primary users and secondary users follow Poisson processes, and their service times follow two distinct arbitrary distributions. The stability condition is insensitive to the service time distributions and implies the maximal throughout of secondary users. Moreover, we extend the stability analysis to the system with outgoing calls. For a special case of exponential service time distributions, we analyze the buffered system in depth to show the effect of parameters on the delay performance and the mean number of interruptions of secondary users. Our simulations for distributions rather than exponential reveal that the mean number of terminations for secondary users is less sensitive to the service time distribution of primary users.

An Analysis of a Stochastic ON-OFF Queueing Mobility Model for Software-Defined Vehicle Networks

We have recently witnessed a number of new software-defined paradigms of VANET in what is referred to as software-defined vehicle networks (SDVN). In order to evaluate the performance of these new proposals and architectures, analytical and simulation models are needed. In this paper, we propose an analytical model based on ON-OFF queueing networks under exponential and general service time distributions. The model can be used to evaluate the performance of SDVNs and takes into account the effect of mobility such as, hand overs, node turning ON/OFF, node going temporary out of coverage, and intermittent connections. This mobility effect was modelled as a queueing station with exponentially random ON-OFF service times, where traffic arrives according to a Poisson random process during the exponentially random ON period and the service time is exponentially distributed. However, during the OFF period the service time is exponentially distributed but with lower rates. We studied the ON-OFF queueing behaviour extensively for both finite-capacity and infinite-capacity queues. Three hypothetical SDVN scenarios were considered, taking into account the effect of mobility and the large number of connected nodes. Results were cross-validated with those obtained by a simulation model. These tools will be valuable for researchers interested in getting quantitative answers for their SDVN architectures.

Optimal pricing in multi server systems

We study optimal service pricing in server farms where customers arrive according to a renewal process and have independent and identical (i.i.d.) exponential service times and i.i.d. valuations of the service. The service provider charges a time varying service fee aiming at maximizing its revenue rate. The customers that find free servers and service fees lesser than their valuation join for the service else they leave without waiting. We consider both finite server and infinite server farms. We solve the optimal pricing problems using the framework of Markov decision problems. We show that the optimal prices depend on the number of free servers. We propose algorithms to compute the optimal prices. We also establish several properties of the optimal prices and the corresponding revenue rates in the case of Poisson customer arrivals. We illustrate all our findings via numerical evaluation.

A Probabilistic Approach to Growth Networks

Single-class closed queueing networks, consisting of infinite-server and single-server queues with exponential service times and probabilistic routing, admit product-from solutions. Such solutions, although seemingly tractable, are difficult to characterize explicitly for practically relevant problems due to the exponential combinatorial complexity of its normalization constant (partition function). In “A Probabilistic Approach to Growth Networks,” Jelenković, Kondev, Mohapatra, and Momčilović develop a novel methodology, based on a probabilistic representation of product-form solutions and large-deviations concentration inequalities, which identifies distinct operating regimes and yields explicit expressions for the marginal distributions of queue lengths. From a methodological perspective, a fundamental feature of the proposed approach is that it provides exact results for order-one probabilities, even though the analysis involves large-deviations rate functions, which characterize only vanishing probabilities on a logarithmic scale.

Processor Sharing Queues With Impatient Customers and State-Dependent Rates

We study queues with impatient customers and Processor Sharing (PS) discipline as well as other variants of PS discipline, namely, Discriminatory Processor Sharing (DPS) and Generalized Processor Sharing (GPS) disciplines, where customers have deadlines until the end of service (DES). Customers arrive according to a state-dependent Poisson process and have general impatience. Customers have exponential service times with state-dependent service rates. Analytical methods based on simple Markov chains are given for the performance analysis of such queues. The principal measures of performance are the steady-state probability of missing deadline and the steady-state probability of blocking. Similar results are obtained for related queues with Random Order Service (ROS) discipline where customers have deadlines until the beginning of service (DBS). In view of a lack of exact analytical results for First Come First Served (FCFS) queues with state-dependent rates, a highly accurate approximation method is also given for these latter queues. The efficacy and accuracy of the approach are illustrated by some numerical examples and simulation experiments.

Age of Information of Parallel Server Systems with Energy Harvesting.

Motivated by current communication networks in which users can choose different transmission channels to operate and also by the recent growth of renewable energy sources, we study the average Age of Information of a status update system that is formed by two parallel homogeneous servers and such that there is an energy source that feeds the system following a random process. An update, after getting service, is delivered to the monitor if there is energy in a battery. However, if the battery is empty, the status update is lost. We allow preemption of updates in service and we assume Poisson generation times of status updates and exponential service times. We show that the average Age of Information can be characterized by solving a system with eight linear equations. Then, we show that, when the arrival rate to both servers is large, the average Age of Information is one divided by the sum of the service rates of the servers. We also perform a numerical analysis to compare the performance of our model with that of a single server with energy harvesting and to study in detail the aforementioned convergence result.

Availability and maintenance modeling of a batch service queuing system

PurposeThe purpose of the paper is to analyze reliability characteristics of batch service queuing system with a single server model that envisages Poisson input process and exponential service times under first come, first served (FCFS) queue discipline.Design/methodology/approachWith the help of renewal theory and stochastic processes, a model has been designed to discuss the reliability and its characteristics.FindingsThe instantaneous and steady-state availability along with the maintenance model of the systems subject to generalized M/Mb/1 queuing model is derived, and a few particular cases for availability are obtained as well. For supporting the developed model, a case study on electrical distribution system (EDS) has been illustrated, which also includes a comparison for the system subject to M/Mb/1 queuing model and the system without any queue (delay).Originality/valueIt is a quite realistic model that may aid to remove congestion in the system while repairing.

On the Time to Buffer Overflow in a Queueing Model with a General Independent Input Stream and Power-Saving Mechanism Based on Working Vacations.

A single server queue with a limited buffer and an energy-saving mechanism based on a single working vacation policy is analyzed. The general independent input stream and exponential service times are considered. When the queue is empty after a service completion epoch, the server lowers the service speed for a random amount of time following an exponential distribution. Packets that arrive while the buffer is saturated are rejected. The analysis is focused on the duration of the time period with no packet losses. A system of equations for the transient time to the first buffer overflow cumulative distribution functions conditioned by the initial state and working mode of the service unit is stated using the idea of an embedded Markov chain and the continuous version of the law of total probability. The explicit representation for the Laplace transform of considered characteristics is found using a linear algebra-based approach. The results are illustrated using numerical examples, and the impact of the key parameters of the model is investigated.

Covert Cycle Stealing in a Single FIFO Server

Consider a setting where Willie generates a Poisson stream of jobs and routes them to a single server that follows the first-in first-out discipline. Suppose there is an adversary Alice, who desires to receive service without being detected. We ask the question: What is the number of jobs that she can receive covertly, i.e., without being detected by Willie? In the case where both Willie and Alice jobs have exponential service times with respective rates μ 1 and μ 2 , we demonstrate a phase-transition when Alice adopts the strategy of inserting a single job probabilistically when the server idles: over n busy periods, she can achieve a covert throughput, measured by the expected number of jobs covertly inserted, of O (√ n ) when μ 1 &lt; 2 μ 2 , O (√ n log n ) when μ 1 = 2μ 2 , and O ( n μ 2 /μ 1 ) when μ 1 &gt; 2μ 2 . When both Willie and Alice jobs have general service times, we establish an upper bound for the number of jobs Alice can execute covertly. This bound is related to the Fisher information. More general insertion policies are also discussed.