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In this article, we introduce a nonparametric (or distribution-free) estimator for traffic intensity in multi-server queues, which has not yet been discussed in the literature. Because this is a very useful model with many potential practical applications, it is the main focus of this study. We compare the performance of a new nonparametric estimator for situations in which the use of Markovian multi-server queues (M/M/s queues in Kendall notation) is adequate or in which it is necessary to consider multi-server queues with general arrival and general service times. We show that, when the parametric Markovian assumptions of M/M/s queues are satisfied, the new estimator is not superior to the maximum likelihood estimator based on the Markovian assumption with respect to M/M/s queues. However, for situations in which the interarrival time distribution and/or the service time distribution cannot be considered exponential (that is, non-Markovian), the new nonparametric estimator is superior. All evaluations are carried out using Monte Carlo simulations. A detailed numerical example is presented to show the usefulness of the technique for practical applications. [ABSTRACT FROM AUTHOR]

A queueing-inventory system where the server is subject to breakdown is examined. The products in inventory are provided by production. Two different types of customers are considered. Each customer type arrives in the system according to a Poisson process with different parameters. There is only one server and the service time follows an exponential distribution. Service may be interrupted randomly due to breakdowns. There may be minor or major breakdowns and after the recovery process, the priority customer class gets service first. The described system is formulated as a level-dependent quasi-birth-death process (LDQBD). The stationarity condition is provided and the steady-state probabilities are calculated. A comprehensive numerical study is presented that examines the sensitivity of the performance measures to the change of system parameters and also investigates the optimal inventory policy according to the considered cost function. [ABSTRACT FROM AUTHOR]

We study a production-inventory system with two customer classes with different priorities which are admitted to the system following a flexible admission control scheme. The inventory management is according to a base stock policy and arriving demand which finds the inventory depleted is lost (lost sales). We analyse the global balance equations of the associated Markov process and derive structural properties of the steady state distribution which provide insights into the equilibrium behaviour of the system. We derive a sufficient condition for ergodicity using the Foster-Lyapunov stability criterion. For a special case we show that the condition is necessary as well. [ABSTRACT FROM AUTHOR]

We analyze a queueing model for cognitive wireless networks using the asymptotic-diffusion method (Moiseev et al. (2020); Nazarov et al. (2020)). Cognitive wireless is a technology that resolves radio spectrum shortages by allowing secondary users (SUs, unlicensed users) to occupy channels initially assigned to primary users (PUs, licensed users). SUs need to sense the channel availability upon arrival. After sensing, an SU can transmit if there is an idle channel; otherwise, the SU must continue sensing. We consider the situation where SUs may be interrupted by the arrivals of PUs when all channels are occupied. We derive a diffusion limit for the queueing model when the SUs' mean sensing time tends to infinity. The diffusion limit leads to an approximate probability distribution of the number of sensing SUs. Finally, we derive a necessary stability condition which turns out to be consistent with the sufficient condition obtained in previous research. [ABSTRACT FROM AUTHOR]

In this study, a continuous review (s,Q) inventory system with a service facility is examined. There is only one server and a limited number of customers waiting rooms in this facility. The demands arrive to the queueing-inventory system according to the Poisson process. Every customer needs a single product with a service period that is unpredictable and distributed arbitrarily. An external supplier replenishes the inventory, and the lead time for the reorder is predicated on an independent exponential distribution. Demands that arise during a stock out period must wait in the waiting area, and when the ordered items arrive, they are served using the first-come-first-serve queueing discipline. With the help of the imbedded Markov chain technique, we are able to compute the joint probability distribution of the number of customers in the system and the number of items in inventory at post-departure epoch. With the remaining service time of a customer in service as the supplementary variable, we are able to relate the system length distributions at post-departure and random epochs in order to determine the joint probability distribution at random epoch. The analysis of waiting time of an accepted customer in the queue is also examined. Several stationary system performance measures are computed and the total expected cost is determined under an appropriate cost structure to determine the optimal values for waiting space (N), reorder level (s), and order quantity (Q). In order to explain the important performance indicators of the system, some numerical findings are given for a variety of model parameters. [ABSTRACT FROM AUTHOR]

We study a supply chain consisting of production-inventory systems at several locations which are coupled by a common supplier. Demand of customers arrives at each production system according to a Poisson process and is lost if the local inventory is depleted ("lost sales"). To satisfy a customer's demand a server at the production system needs raw material from the associated local inventory. The supplier manufactures raw material to replenish the local inventories, which are controlled by a continuous review base stock policy. The routing of items depends on the on-hand inventory at the locations with the aim to obtain "load balancing". We show that the stationary distribution has a product form of the marginal distributions of the production subsystem and the inventory-replenishment subsystem. For the marginal distribution of the production subsystem we derive an explicit solution and for the marginal distribution of the inventory-replenishment subsystem we deduce an explicit solution or a recursive algorithm for some special cases. [ABSTRACT FROM AUTHOR]

A network of queueing-inventory systems is considered where the inventories are replenished by a common central server. Travel times for transport of the items send out from the center and an adaptive dispatching regime to direct transportation towards the locations with queueing-inventory systems are incorporated in the model. The stationary distribution of the system is of product form and is explicitly given. An optimization procedure is developed to find for given locations of the queueing-inventory systems an optimal location for the replenishment center. Optimization is with respect to overall utilization of the resources measured in total throughput of the queueing-inventory systems. [ABSTRACT FROM AUTHOR]

In the recent strategic queueing literature, there is a large number of papers that study the join-or-balk dilemma in queueing systems with server's on-off periods, modeling vacations and failures. These studies consider the customers as discrete units and adopt the assumption that reneging is not permitted. In the present paper, we depart from this framework and study the effect of the reneging option in such systems. We consider the fluid on-off model of the basic queue with vacations/failures and study reneging vs. no-reneging when customers are strategic. We derive the equilibrium customer strategies and the corresponding performance measures of the system, and we use them to study the equilibrium throughput and social welfare. The main finding is that the existence of the reneging option is very beneficial for overloaded systems, i.e., for such systems balking alone is not sufficient to achieve good outcomes. On the contrary, for underloaded systems the reneging option is not particularly valuable. [ABSTRACT FROM AUTHOR]

A two-commodity queueing-inventory system with phase-type service times and exponential lead times is considered. There are two types of customers; Type 1 and Type 2. Demands from each customer type occur independently according to a Poisson process with different rates whereas the service times follow a phase-type distribution. Type 1 customers have a non-preemptive priority over Type 2 customers. We assume a finite waiting space for Type 1 customers whereas there is no limit on the waiting room for Type 2 customers. Type i customers demand only commodity i, i = 1 , 2 . For the ith commodity, S i and s i represent, respectively, the maximum inventory level and the reorder level. Whenever the inventory level of ith commodity drops to s i , an order is placed from retailer-i to make the inventory level S i . The lead times of the commodities are exponentially distributed with different parameters. When there is a Type i customer waiting in the queue, if the inventory level of ith commodity is zero (or reaches zero), a decision of immediate purchase is made so as not to lose the waiting customer. The queueing-inventory model in the steady-state is analyzed using the matrix-geometric method. The system performance is examined for different values of parameters. Besides, an optimization study is performed for some system parameters. [ABSTRACT FROM AUTHOR]

This paper analyses the M/M(a,b)/1/MWV queuing model with various types of breakdowns. During peak times, the system has breakdown due to server unavailability. The model considers system with two types of breakdowns. The system may breakdown in two different ways during busy and working vacation stages. Customer arrives to the system with parameter λv which follows Poisson distribution and server provides service in regular busy period with parameter µrb and under the multiple working vacations it provides service with parameter µwv with exponential distribution. In this model batches of customers are served under General Bulk Service Rule. The system may breakdown at any time. The breakdown that occurs during working vacation is denoted as βv1 and the breakdown during busy state is denoted as βv2. The steady-state equation, the performance of measures for the system and particular cases of described model are derived. Finally, a real life example demonstrated the validity of the model in a proper way. [ABSTRACT FROM AUTHOR]

We consider the following network model motivated, in particular, by blockchains and peer-to-peer live streaming. Data packet flows arrive at the network nodes and need to be disseminated to all other nodes. Packets are relayed through the network via links of finite capacity. A packet leaves the network when it is disseminated to all nodes. Our focus is on two communication disciplines, which determine the order in which packets are transmitted over each link, namely Random-Useful (RU) and Oldest-Useful (OU). We show that RU has the maximum stability region in a general network. For the OU we demonstrate that, somewhat surprisingly, it does not in general have the maximum stability region. We prove that OU does achieve maximum stability in the important special case of a symmetric network, given by the full graph with equal capacities on all links and equal arrival rates at all nodes. We also give other stability results, and compare different disciplines' performances in a symmetric system via simulation. Finally, we study the cumulative delays experienced by a packet as it propagates through the symmetric system, specifically the delay asymptotic behavior as N → ∞ . We put forward some conjectures about this behavior, supported by heuristic arguments and simulation experiments. [ABSTRACT FROM AUTHOR]

Various load balancing policies are known to achieve vanishing waiting times in the large-scale limit, that is, when the number of servers tends to infinity. These policies either require a communication overhead of one message per job or require job size information. Load balancing policies with an overhead below one message per job are called hyperscalable policies. While these policies often have bounded queue length in the large-scale limit and work well when the overhead is somewhat below one, they show poor performance when the communication overhead becomes small, that is, the mean response time tends to infinity when the overhead tends to zero even at low loads. In this paper, we introduce a hyperscalable load balancing policy, called Join-Up-To(m), that remains effective even when the communication overhead tends to zero. To study its performance under general job size distributions, we make use of the "queue at the cavity" approach. We provide explicit results for the first two moments of the response time, the generating function of the queue length distribution and the Laplace transform of the response time. These results show that the mean response time only depends on the first two moments of the job size distribution. [ABSTRACT FROM AUTHOR]

This paper considers an M/G/1 repairable queueing system with vacations and patient server. It is assumed that the service station may break down during service period. Whenever a busy period ends, the server is allowed to take a vacation. When the server returns from vacation and finds an idle system, he/she waits for a patience period within the system. If a customer arrives during this period, a new busy period starts at once. Otherwise, a new vacation begins immediately at the completion epoch of the patience period. Using the total probability decomposition technique and the Laplace transform tool, some crucial reliability indices of the service station are derived, including the distribution of the time to first failure, the transient and steady-state unavailability, the mean number of failures during time interval (0, t] and the steady-state failure frequency. The results of our study indicate that the reliability measures obtained satisfy the stochastic decomposition property. Finally, some numerical examples are provided to illustrate the influence of various system parameters on these reliability indices. [ABSTRACT FROM AUTHOR]

Waiting is a major factor influencing the perception of delay-sensitive customers in the service industry. In the process of queueing, some customers often have a psychological expectation of waiting time in the face of uncertain delay information, so that customer service utility depends not only on the actual waiting time, but also on the relative amount of the actual waiting time and the psychological expectation of waiting time. Therefore, this paper investigates how the reference time effect affects heterogeneous customers' queueing decisions and service system efficiency measures (system throughput and social welfare) in an M/G/1 queue with limited service resources and capacity. The results show that the equilibrium joining probability of customers, the system throughput and social welfare are relatively higher as the proportion of customers with high tolerance levels in the queue increases. In addition, the maintenance of customer homogeneity is better for the improvement of service resource utilization, while the maintenance of customer heterogeneity is better for social welfare. As the psychological expected waiting time increases, the equilibrium joining probability of potential customers and the system throughput increase, while the equilibrium joining probability of existing customers decreases, and the social welfare shows a non-monotonic trend of first decreasing and then increasing. The equilibrium queueing strategies for each type of customer and the service system efficiency measures are not monotonic with the change of the reference time effect parameter. Finally, the optimal social welfare is increasing with respect to the degree of reference time effect and the psychological expectation of waiting time. [ABSTRACT FROM AUTHOR]

This work aims to study customer strategic behavior and operational optimization of an M/M/1 queue with N-policy and dynamic service-rate control in general, and of a make-to-order production system with a strategic manufacturer as a practical example in particular. With the N-policy and hysteretic control, the service rate is dynamically adjusted according to the queue length. An empty system is required to initiate service only when the number of waiting customers reaches a given threshold N. If the queue length exceeds M (> N) , the server serves customers with a higher service rate until the number of customers is lower than another threshold m. Based on a linear reward-cost structure, arriving customers are allowed to decide whether to join the system or not. The focus is on comparing the self equilibrium performance and socially-optimal arrival under the interaction between the server’s states and customers’ joining decisions. The influence of N-policy and hysteretic control on customer’s equilibrium joining probability under unobservable scenario is analysed. The optimal value of threshold N and m are discussed by taking server’s operational cost into consideration. These findings have important managerial implications on the overall social welfare with dynamic service rate. [ABSTRACT FROM AUTHOR]

In communication networks, the signals from mobile phones are transmitted through Base Stations(BS). Nowadays the usage of mobile phones has tremendously increased. Approximately 80% of the input energy is dissipated as heat by the components of base stations (Holtkamp et al. in IEEE J Sel Areas Commun PP(99):1–10, 2013; Wu et al. in IEEE Commun Surveys Tuts 17(2):803–826, 2015). Various strategies are available to minimize the power consumption in a base station. One of the effective ways to reduce power consumption is sleeping strategy. Some of them are single sleep, multiple sleep, light sleep, deep sleep, N-policy. According to N-policy, the BS only enters the active state when there are N URs waiting in the queue. In this paper, a base station is modelled as finite buffer M/G/1 queue with two different sleeping modes namely, short sleep and long sleep. The probability generating function of queue size distribution is derived using supplementary variable technique. The expressions for expected power consumption and delay are derived. The proposed work is numerically justified using simulation. The simulation results are obtained and graphically verified in the proposed work. [ABSTRACT FROM AUTHOR]

We study finite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. We show that the particle jump rates determine explicitly a unique partition of the system into maximal stable sub-systems, and that this partition can be obtained by a linear-time algorithm using only elementary arithmetic. The internal configuration of each stable sub-system possesses an explicit product-geometric limiting distribution, and the location of each stable sub-system obeys a strong law of large numbers with an explicit speed; the characteristic parameters of each stable sub-system are simple functions of the rate parameters for the corresponding particles. For the case where the entire system is stable, we provide a central limit theorem describing the fluctuations around the law of large numbers. Our approach draws on ramifications, in the exclusion context, of classical work of Goodman and Massey on partially-stable Jackson queueing networks. [ABSTRACT FROM AUTHOR]

In this paper, we incorporate loss preference into an M/M/1 queueing with a threshold disclosure policy and analyze its impact on the customers’ queueing strategies and the queueing system’s idle stationary probability. In the queueing system, customers are strategic and divided into two groups: the informed and the uninformed. Informed customers are assumed to be fully rational, whereas uninformed customers are assumed to have loss preference. Uninformed customers with loss preference are categorized into two types according to their asymmetry perceptions, which anchor on the difference between gain and loss: loss neutrality and loss aversion. We firstly determine customers’ equilibrium decisions, and then derive the idle stationary probability at equilibrium. We find that loss preference reduces the customers’ joining probability, and results in a higher idle stationary probability. Furthermore, we find that for the uninformed customers with stronger loss aversion, the system manager should lower the threshold of disclosure to maintain a stable demand of uninformed customers. In addition, in the case of mixed-strategy at equilibrium, with the increase of the threshold of disclosure, the idle stationary probability increases for an underloaded queue. However, for an overloaded queue, the idle stationary probability decreases with increasing the threshold of disclosure. [ABSTRACT FROM AUTHOR]

We study three non-equivalent queueing models in continuous time that each generalise the classical M/M/1 queue in a different way. Inter-event times in all models are Mittag-Leffler distributed, which is a heavy tail distribution with no moments. For each of the models we answer the question of the queue being at zero infinitely often (the 'recurrence' regime) or not (the transient regime). Aside from this question, the different analytical properties of each models allow us to answer a number of questions such as existence and description of equilibrium distributions, mixing times, asymptotic behaviour of return probabilities and moments and functional limit theorems. [ABSTRACT FROM AUTHOR]

We consider the optimal scheduling problem in the M/G/1 queue. While this is a thoroughly studied problem when the target is to minimize the mean delay, there are still open questions related to some other objective functions. In this paper, we focus on minimizing mean slowdown among non-anticipating polices, which may utilize the attained service of the jobs but not their remaining service time when making scheduling decisions. By applying the Gittins index approach, we give necessary and sufficient conditions for the jobs' service time distribution under which the well-known scheduling policies first come first served and foreground background are optimal with respect to the mean slowdown. Furthermore, we characterize the optimal non-anticipating policy in the multi-class case for certain types of service time distributions. In fact, our results cover a more general objective function than just the mean slowdown, since we allow the holding costs for a job to depend on its own service time S via a generic function c(S). When minimizing the mean slowdown, this function reads as c (x) = 1 / x . [ABSTRACT FROM AUTHOR]

Consider a stable M/G/1 system in which, at time t = 0 , there are exactly n customers with residual service times equal to v 1 , v 2 , ... , v n . In addition, assume that there is an extra customer c who arrives at time t = 0 and has a service requirement of x. The externalities which are created by c are equal to the total waiting time that others will save if her service requirement is reduced to zero. In this work, we study the joint distribution (parameterized by n , v 1 , v 2 , ... , v n , x ) of the externalities created by c when the underlying service distribution is either last-come, first-served with preemption or first-come, first-served. We start by proving a decomposition of the externalities under the above-mentioned service disciplines. Then, this decomposition is used to derive several other results regarding the externalities: moments, asymptotic approximations as x → ∞ , asymptotics of the tail distribution, and a functional central limit theorem. [ABSTRACT FROM AUTHOR]

The impact of information about the quality of service on marketing can be demonstrated by a competition between an observable queue and an unobservable queue. An illustration was given by an earlier paper of Hassin (Manag Sci 42:618–623, 1996), which considers two gas stations located one after the other on a main road. A driver who needs to fill the gas can see the queue length upon arriving at the first station, but not the second one. The driver decides which station to enter based only on this partial information for minimizing the expected waiting time. Via extensive numerical experiments, Hassin showed that the first server enjoys a larger market share, which led him to making a number of challenging conjectures. In this paper, we prove that all the numerical observations and conjectures are true so that the advantage of being the first server is assured. [ABSTRACT FROM AUTHOR]

This study considers a multi-class single-server retrial queue with constant retrial rates in which service times are generally distributed with different service rates for each class. Avrachenkov (Queueing Systems 100:177–179, 2022) formulated the conjecture on the stability of this queueing system. We completely resolve the conjecture by using the fluid limit approach. [ABSTRACT FROM AUTHOR]

This paper deals with testing traffic intensity, which is the most important parameter of the M / M / 1 queue system. The Wald-type sequential probability ratio test (SPRT) is performed to determine traffic intensity by taking advantage of the fact that the total number of customers arriving up to the kth service period, including the kth service period, has a negative binomial distribution. Acceptance and rejection limits obtained from the SPRT are arranged as a multiple sampling plan. This plan is applied with a finite Markov chain. Then, the Operating Characteristic (OC) function and the Average Sample Number (ASN) are obtained precisely. Obtaining the standard errors of the sample number compared to the studies on SPRT applications is the novelty brought by this paper. The sequential analysis method based on the finite Markov chain is also applied to a numerical sample. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analysis of an infinite-capacity batch arrival multi-server queueing system with Bernoulli feedback, synchronous multiple and single working vacation policies, waiting servers, reneging and retention of reneged customers. The steady-state solution of the queueing system is obtained by using probability generating function (PGF). In addition, important performance measures of the queueing system are derived. Then, a cost model is formulated in order to carry out the parameter optimization using genetic algorithm (GA). Finally, numerical study is presented in which various system performance measures are evaluated based on supposed numerical values given to the system parameters.

In the literature, retrial queues with batch arrivals and heavy-tailed service times have been studied and the so-called equivalence theorem has been established under the condition that the service time is heavier than the batch size. The equivalence theorem provides the distribution (or tail) equivalence between the total number of customers in the system for the retrial queue and the total number of customers in the corresponding standard (non-retrial) queue. In this paper, under the assumption of regularly varying tails, we eliminate this condition by allowing that the service time can be either heavier or lighter than the batch size. The main contribution made in this paper is an asymptotic characterization of the difference between two tail probabilities: the probability of the total number of customers in the system for the M X / G / 1 retrial queue and the probability of the total number of customers in the corresponding standard (non-retrial) queue. [ABSTRACT FROM AUTHOR]

This paper considers an m-station cascade system with renewal arrival processes and general service time distributions. Miyazawa and Morozov (Queueing Syst 100:225–227, 2022a; Stability of a cascade system with two stations and its extension for multiple stations, 2022b. arXiv:2203.14294v1) formulated a conjecture on the stability of this cascade system. In this paper, we show that this conjecture is not true for m ≥ 3 . We also show that when m ≥ 3 , the stability condition is not determined solely by the moments of the interarrival and service times, but also depends on their distributions. In addition, we provide the necessary and sufficient condition for the stability of this cascade system by modifying a result of Miyazawa and Morozov (2022). [ABSTRACT FROM AUTHOR]

In this paper, we analyse a queueing model with two types of requests arriving in a marked Markov arrival process. Type-1 requests require a constant service rate, while type-2 requests admit a flexible service rate. Mixed service discipline is considered. It is defined as follows. The number of type-1 requests that can be processed by the system simultaneously is restricted. Type-2 requests receive service according to the classical processor sharing discipline and use all currently available (not occupied by type-1 requests) system bandwidth. Type-2 requests can be impatient and leave the system without receiving complete service. The system behavior is described by a multidimensional Markov chain. The infinitesimal generator of this chain is derived. The transparent ergodicity condition is obtained, and the stationary performance measures of the system are computed. A numerical example is presented, including consideration of the problem of choosing the optimal values of the system bandwidth and its share dedicated to the service of type-1 requests. [ABSTRACT FROM AUTHOR]

We consider a non-preemptive priority M/M/1 retrial queue with two classes of customers (low-priority and high-priority customers) and delayed vacations. When the server is unavailable, an arriving high-priority customer can wait in line, whereas an arriving low-priority customer needs to enter a virtual queue and retry later. After completing a service, the server will remain idle for a reserved idle time if it finds no high-priority customers in the system. Arrivals during the reserved idle period will be served immediately. Otherwise, if no customers arrive during this interval, the server will switch to the vacation state. By constructing a three-dimensional Markov chain, we successively obtain the stability condition of the system and some main performance measures. Then depending on a linear reward-cost structure, we derive customers’ two-dimensional equilibrium joining strategies in the fully unobservable case. Due to the complexity of the social welfare function, we explore the socially optimal joining strategies through the Particle Swarm Optimization (PSO) algorithm. Finally, we illustrate the impact of system parameters on the two types of joining strategies via numerical experiments. [ABSTRACT FROM AUTHOR]

Reducing energy consumption is the vital goal of green communication. Base station (BS) is a radio receiver/transmitter that serves as the hub of the local wireless network. It is a gateway between a wired network and the wireless network. BS consumes high energy to receive and transfer the signals. Power consumption in base station can be minimized by using effective sleep and wake-up/setup operations with a tolerable delay. In this research work, the service process of the BS is considered as an M/G/1 queue with close down, sleep and setup. The strategy N-Policy is introduced to awake the BS from multiple sleeps (MS) after a predefined number N of user requests (URs) accumulated in the system. The supplementary variable technique is used to obtain the probability-generating functions and the steady-state probabilities for different states of the BS. The mean delay of the UR and mean power consumption of the BS are also derived. Also, the comparative analysis of the proposed model with the existing model has been presented. Computational results show that multiple sleeps with N-policy consumes less power than multiple sleeps without N-policy. [ABSTRACT FROM AUTHOR]

Software application when executed on computer systems having disparate hardware features, result in dissimilar performance and power. Therefore, selecting systems with optimal performance and power values for application execution is an essential problem to address. However, access to many physical systems is required to collect performance power, which is a demanding task. Therefore, our first objective is to build an accurate prediction model for physical systems to achieve the second objective of computer system selection. To achieve the first objective, we propose a novel model, "cross performance and power prediction with scaling." We develop a cross prediction model for physical systems by training a decision tree machine learning algorithm on performance and power datasets obtained from a large number of simulation systems built in the Gem5 simulator using emulation mode. However, design differences between Gem5 systems and physical systems lead to large prediction inaccuracies. We determine the application-specific "scaling factor" to compensate for the prediction inaccuracies and apply it to the predicted values for accurate physical systems predictions. We evaluate our model on well-known applications from SD-VBS and MiBench benchmarks achieving errors of 10–25% and 6–40% for performance and power for general-purpose systems. With accurate predictions for physical systems from our model, we achieve the second goal of computer system selection. [ABSTRACT FROM AUTHOR]

We analyse a generalisation of the stochastic gene expression model studied recently in Fromion et al. (SIAM J Appl Math 73:195–211, 2013) and Robert (Probab Surv 16:277–332, 2019) that keeps track of the production of both mRNA and protein molecules, using techniques from the theory of point processes, as well as ideas from the theory of matrix-analytic methods. Here, both the activity of a gene and the creation of mRNA are modelled with an arbitrary Markovian Arrival Process governed by finitely many phases, and each mRNA molecule during its lifetime gives rise to protein molecules in accordance with a Poisson process. This modification is important, as Markovian Arrival Processes can be used to approximate many types of point processes on the nonnegative real line, meaning this framework allows us to further relax our assumptions on the overall process of transcription. [ABSTRACT FROM AUTHOR]

We establish new nonuniform estimates of the convergence rate to the exponential distribution in the boundary theorem for geometric sums. We give examples of their application to extrema of regenerative random birth and death processes. [ABSTRACT FROM AUTHOR]

We consider the M/D/1 retrial queueing system with constant retrial times, which makes up a natural abstraction for optical fibre delay line buffers. Drawing on a time-discretisation approach and on an equivalence with polling systems, we find explicit expressions for the distribution of the number of retrials, and the probability generating function of the number of customers in orbit. While the state space of the queueing system at hand is complicated, the results are strikingly simple. The number of retrials follows a geometric distribution, while the orbit size decomposes into two independent random variables: the system content of the M/D/1 queue at departure times and the orbit size of the M/D/1 retrial queue when the server is idle. We finally obtain explicit expressions for the retrial rate after a departure and for the distribution of the time until the nth retrial after a departure. [ABSTRACT FROM AUTHOR]

This paper addresses the ubiquity of remarkable measures on graphs and their applications. In many queueing systems, it is necessary to take into account the compatibility constraints between users, or between supplies and demands, and so on. The stability region of such systems can then be seen as a set of measures on graphs, where the measures under consideration represent the arrival flows to the various classes of users, supplies, demands, etc., and the graph represents the compatibilities between those classes. In this paper, we show that these 'stabilizing' measures can always be easily constructed as a simple function of a family of weights on the edges of the graph. Second, we show that the latter measures always coincide with invariant measures of random walks on the graph under consideration. Some arguments in the proofs rely on the so-called matching rates of specific stochastic matching models. As a by-product of these arguments, we show that, in several cases, the matching rates are independent of the matching policy, that is, the rule for choosing a match between various compatible elements. [ABSTRACT FROM AUTHOR]

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Toward that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an M/M/2 queueing system with a second optional service that operates under a triadic (0, Q, N, M) policy. The stability condition and stationary probability distribution of the system size are computed by adopting the matrix-geometric method. Numerous system performance measures are also derived. The quadratic fit search method is applied to the cost optimization problem in order to optimize service rates during the first essential service and the second optional service. Finally, numerical results are presented for the illustration of the effect of model parameters. [ABSTRACT FROM AUTHOR]

This paper studies a discrete-time single server batch arrival and batch service queueing model with unlimited waiting space. The discrete-time batch Markovian arrival process and discrete-time batch Markovian service process, respectively, manage the arrival and service processes. We adopt the UL-type RG-factorization approach based on censoring methodology for variable size batch service queue to calculate the stationary probability vectors of the transition probability matrix with general structure Markov chain at outside observer’s epoch. We reblock the transition probability matrix to its desired M/G/1 structure to find the stationary probability vectors at outside observer’s epoch for fixed size batch service queue using the matrix analytic method. We also develop relationships to determine probability vector expressions for other important time epochs such as pre-arrival, intermediate, post-departure, and random epochs. The most challenging aspect of our study is to obtain the probability mass functions of sojourn time in the system for both the variable and fixed size batch service queues. We use our suggested queueing model to derive the results of several specific well-known queueing models. We also discuss about possible managerial implication of our model to produce fruit juices in manufacturing industry. We present computational experience based on the execution of parametrized experiments with various categories in order to validate the analytical results reported in this study. [ABSTRACT FROM AUTHOR]

In this paper, we consider a double-ended queue with First-Come-First-Match discipline (also known as matched queues) under customers’ flexible and impatient behaviors. Such a system can be expressed as a level-dependent quasi-birth-and-death (QBD) process with infinitely many phases. The stability condition of the queueing system is given by using the mean drift technique. To deal with the level-dependent QBD process, we apply the RG-factorizations to obtain stationary probability vectors. Based on this, the queue size distributions and the average stationary queue lengths are given. Furthermore, we provide an effective method to discuss the sojourn time of any arriving customer and to compute the average sojourn time by using the technique of the first passage times and the phase-type (PH) distributions. Finally, some numerical examples are employed to illustrate how the performance measures are influenced by key system parameters. [ABSTRACT FROM AUTHOR]

We consider a random walk in Z 2 and establish a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper directed state-crossings and different sets of states crossings. As well, we extend the results to d-dimensional random walks, d ≥ 2 , in bounded areas. [ABSTRACT FROM AUTHOR]