The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are $r$ server stations. At the initial time moment all units are distributed in the server stations, and the $i$th server station contains $N_i$ units, $i=1,2,...,r$, where all the values $N_i$ are large numbers of the same order. The total number of client stations is equal to $k$. The expected times between departures in the client stations are small values of the order $O(N^{-1})$ ~ $(N=N_1+N_2+...+N_r)$. After service completion in the $i$th server station a unit is transmitted to the $j$th client station with probability $p_{i,j}$ ~ ($j=1,2,...,k$), and being served in the $j$th client station the unit returns to the $i$th server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.
This paper considers a discrete-time single-server queue with a single acceptance period for a Poissonian population of homogeneous customers. Customers are served on a first-come first-served (FCFS) basis, and their service times are independent and identically distributed with a general distribution. We assume that each customer chooses her/his arrival-time slot with the goal of minimizing her/his expected waiting time in competition with other customers. For this queueing game, we derive a symmetric (mixed-strategy) Nash equilibrium; that is, an equilibrium arrival-time distribution of homo…
A single server retrial queueing system with two-classes of orbiting customers, and general class dependent service times is considered. If an arriving customer finds the server unavailable, it enters a virtual queue, called the orbit, according to its type. The customers from the orbits retry independently to access the server according to the constant retrial policy. We derive the generating function of the stationary distribution of the number of orbiting customers at service completion epochs in terms of the solution of a Riemann boundary value problem. For the symmetrical system we also d…
This paper studies an infinite buffer single server queueing model with exponentially distributed service times and negative arrivals. The ordinary (positive) customers arrive in batches of random size according to renewal arrival process, and joins the queue/server for service. The negative arrivals are characterized by two independent Poisson arrival processes, a negative customer which removes the positive customer undergoing service, if any, and a disaster which makes the system empty by simultaneously removing all the positive customers present in the system. Using the supplementary varia…
The paper studies a multiserver retrial queueing system with $m$ servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter $μ_1$. A time between retrials is exponentially distributed with parameter $μ_2$ for each customer. Using a martingale ap…
This paper examines the performance of multi-class multi-server bipartite queueing systems under a FCFS-ALIS service discipline, where each arriving customer is only compatible with a subset of servers. We analyze the system under conventional heavy-traffic conditions, where the traffic intensity approaches one from below. Building upon the formulation and results of Afeche et al. 2021, we generalize the model by allowing the vector of arrival rates to approach the heavy-traffic limit from an arbitrary direction. We characterize the steady-state waiting times of the various customer classes an…
This paper introduces and analyzes the notion of throughput suboptimality for many-server queueing systems in heavy traffic. The queueing model under consideration has multiple customer classes, indexed by a finite set $\mathcal{I}$, and heterogenous, exponential servers. Servers are dynamically chosen to serve customers, and buffers are available for customers waiting to be served. The arrival rates and the number of servers are scaled up in such a way that the processes representing the number of class-$i$ customers in the system, $i\in\mathcal{I}$, fluctuate about a static fluid model, that…
The interest to retrial queueing systems is due to their application to telephone systems. The paper studies multiserver retrial queueing systems with $n$ servers. Arrival process is a quite general point process. An arriving customer occupies one of free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after random time retries more and more to occupy a server. The orbit has one waiting space only, and arriving customer, who finds all servers busy and the waiting space occupied, losses from the system. Time intervals between possible retrial…
We consider a semi-open queueing network (SOQN), where a customer requires exactly one resource from the resource pool for service. If there is a resource available, the customer is immediately served and the resource enters an inner network. If there is no resource available, the new customer has to wait in an external queue until one becomes available ("backordering"). When a resource exits the inner network, it is returned to the resource pool and waits for another customer. In this paper, we present a new solution approach. To approximate the inner network with the resource pool of the SOQ…
Strategic customer behavior is strongly influenced by the level of information that is provided to customers. Hence, to optimize the design of queueing systems, many studies consider various versions of the same service model and compare them under different information structures. In particular, two extreme versions are usually considered and compared: the observable in which customers are informed about the number of customers in the system and the unobservable in which they are only informed about the system parameters, e.g., arrival and service rates. In the present work, we study a model …
In the present paper the infinite-server MMAPkGk queueing model with random resource vector of customers, marked MAP arrival and semi-Markov (SM) arrival of catastrophes is considered. The joint generating functions (PGF) of transient and stationary distributions of number of busy servers and numbers of different types served customers, as well as Laplace transformations (LT) of joint distributions of total accumulated resources in the model at moment and total accumulated resources of served customers during time interval are found. The basic differential and renewal equations for transient a…
The infinite-server queueing models with homogeneous and non-homogeneous arrivals of customers and catastrophes are considered. The probability generating functions of joint distributions of numbers of busy servers and served customers, as well as the Laplace-Stieltjes Transforms of distribution of busy period and distribution of busy cycle for the models are found.
This paper investigates a partially observable queueing system with $N$ nodes in which each node has a dedicated arrival stream. There is an extra arrival stream to balance the load of the system by routing its customers to the shortest queue. In addition, a reward-cost structure is considered to analyze customers' strategic behaviours. The equilibrium and socially optimal strategies are derived for the partially observable mean field limit model. Then, we show that the strategies obtained from the mean field model are good approximations to the model with finite $N$ nodes. Finally, numerical …
In this paper, we study a controllable tandem queueing system consisting of two nodes and a controller, in which customers arrive according to a Poisson process and must receive service at both nodes before leaving the system. A decision maker dynamically allocates the number of service resource to each node facility according to the number of customers in each node. In the model, the objective is to minimize the long-run average costs. We cast these problems as Markov decision problems by dynamic programming approach and derive the monotonicity of the optimal allocation policy and the relatio…
When an explicit expression for a probability distribution function $F(x)$ can not be found, asymptotic properties of the tail probability function $\bar{F}(x)=1-F(x)$ are very valuable, since they provide approximations or bounds for system performance, and approaches for computing probability distribution. In this paper, we study tail asymptotic properties for the number of the customers in the $M/G/1$ retrial queueing system. For queueing systems, studies on asymptotic tails have mainly concentrated on the first order asymptotic behavior. To best our knowledge, there is no second order tail…
We study a system, where a random flow of customers is served by servers (called agents) invited on-demand. Each invited agent arrives into the system after a random time; after each service completion, an agent returns to the system or leaves it with some fixed probabilities. Customers and/or agents may be impatient, that is, while waiting in queue, they leave the system at a certain rate (which may be zero). We consider the queue-length-based feedback scheme, which controls the number of pending agent invitations, depending on the customer and agent queue lengths and their changes. The basic…
This paper studies a diffusion model that arises as the limit of a queueing system scheduling problem in the asymptotic heavy traffic regime of Halfin and Whitt. The queueing system consists of several customer classes and many servers working in parallel, grouped in several stations. Servers in different stations offer service to customers of each class at possibly different rates. The control corresponds to selecting what customer class each server serves at each time. The diffusion control problem does not seem to have explicit solutions and therefore a characterization of optimal solutions…
In this paper the infinite server queue model in semi-Markov random environment with k Markov arrival streams, random resources of customers, and catastrophes is considered. After catastrophes occur, all customers in the model are flashed out and the system jumps into recovery station. After the recovery time the model works from the empty state. The transient and stationary joint distributions of numbers of different types of customers in the model at moment t, numbers of different types of served in some interval customers, volume of accumulated resources in the model at moment t, and total …
We analyze the behavior of closed product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route (or class). First, we focus on the stationary behavior and prove the conjecture that the stationary distribution at non-bottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are determined by the solution of a strictly concave optimization problem. Then, we focus on the transie…
The performance of non-preemptive M/M/1 queueing system with two priority is analyzed. By using complementary variable method to make vector Markov process and analyzing the state-change equations of the queueing system, the generating function of two kinds of customers'length distribution are derived under non-preemptive priority .Through further discussion, the probability of the server that it is working or free and average length of two kinds of customers are also derived.
This paper studies the equilibrium behavior of customers in the Geo/Geo/1 queueing system with multiple working vacations. The arriving customers decide whether to join or to balk the queueing systems based on the information of the queue length and the states of the server. In observable queues, partially observable queues and unobservable queues three cases we obtain the equilibrium balking strategies based on the reward-cost structure and socially optimal strategies for all customers. Furthermore, we present some numerical experiments to illustrate the effect of the information level on the…
This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The r…
We consider three parallel service models in which customers of several types are served by several types of servers subject to a bipartite compatibility graph, and the service policy is first come first served. Two of the models have a fixed set of servers. The first is a queueing model in which arriving customers are assigned to the longest idling compatible server if available, or else queue up in a single queue, and servers that become available pick the longest waiting compatible customer, as studied by Adan and Weiss, 2014. The second is a redundancy service model where arriving customer…
In this thesis, we study the queueing systems with heterogeneous servers and service rate uncertainty under the Halfin-Whitt heavy traffic regime. First, we analyse many server queues with abandonments when service rates are i.i.d. random variables. We derive a diffusion approximation using a novel method. The diffusion has a random drift, and hence depending on the realisations of service rates, the system can be in Quality Driven (QD), Efficiency Driven (ED) or Quality-Efficiency-Driven (QED) regime. When the system is under QD or QED regime, the abandonments are negligible in the fluid limi…
Motivated by the ongoing COVID-19 pandemic, this paper investigates customers' infection risk by evaluating the overlapping time of a virtual customer with others in queueing systems. Most of the current methodologies focus on characterizing the risk in stationary systems, which may not apply to the more practical time-varying systems. As such, we propose an approximation framework that relies on the fluid limit to compute the expected overlapping time in time-varying queueing systems. Simulation experiments verify the accuracy of our approach.
The Erlang A model--an M/M/s queue with exponential abandonment--is often used to represent a service system with impatient customers. For this system, the popular square-root staffing rule determines the necessary staffing level to achieve the desirable QED (quality-and-efficiency-driven) service regime; however, the rule also implies that properties of large systems are highly sensitive to parameters. We reveal that the origin of this high sensitivity is due to the operation of large systems at a point of singularity in a phase diagram of service regimes. We can avoid this singularity by imp…
This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-de…
In this paper we models and studies a general vacation queueing model with impatient customers. We first propose a sufficient condition for the existence of the stationary workload process. We then give an integral equation for the independent and identically distributed case. This integral equation is solved when customers arrive according to a Poisson Point process. A relationship between the tail of the waiting time distribution and the tail of service distribution is also given.
This paper studies a closed queueing network containing a hub (a state dependent queueing system with service depending on the number of units residing here) and $k$ satellite stations, which are $GI/M/1$ queueing systems. The number of units in the system, $N$, is assumed to be large. After service completion in the hub, a unit visits a satellite station $j$, $1\leq j\leq k$, with probability $p_j$, and, after the service completion there, returns to the hub. The parameters of service times in the satellite stations and in the hub are proportional to $\frac{1}{N}$. One of the satellite statio…
We develop a robust queueing network analyzer algorithm to approximate the steady-state performance of a single-class open queueing network of single-server queues with Markovian routing. The algorithm allows non-renewal external arrival processes, general service-time distributions and customer feedback. We focus on the customer flows, defined as the continuous-time processes counting customers flowing into or out of the network, or flowing from one queue to another. Each flow is partially characterized by its rate and a continuous function that measures the stochastic variability over time. …
Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: the retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. For this type of system, we propose a detailed stochastic decomposition approach to study its asymptotic behaviour of the tail probability of the number of customers in the steady-state for retrial queues with two types (Type-1 and Type-2) of customers, in …
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 loca…
We consider an $M/M/1$ queueing system with impatient customers with multiple and single vacations. It is assumed that customers are impatient whenever the state of the server. We derive the probability generating functions of the number of customers in the system and we obtain some performance measures.
We consider a heterogeneous queueing system consisting of one large pool of $O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfin-Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where $a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue $Q^r(\infty)$. While there is a rich literature on deriving p…
This paper studies the heavy-traffic (HT) behaviour of queueing networks with a single roving server. External customers arrive at the queues according to independent renewal processes and after completing service, a customer either leaves the system or is routed to another queue. This type of customer routing in queueing networks arises very naturally in many application areas (in production systems, computer- and communication networks, maintenance, etc.). In these networks, the single most important characteristic of the system performance is oftentimes the path time, i.e. the total time sp…
The main contribution of this paper is to present a new sufficient condition for the subexponential asymptotics of the stationary distribution of a GI/GI/1-type Markov chain without jumps from level "infinity" to level zero. For simplicity, we call such Markov chains {\it GI/GI/1-type Markov chains without disasters} because they are often used to analyze semi-Markovian queues without "disasters", which are negative customers who remove all the customers in the system (including themselves) on their arrivals. In this paper, we demonstrate the application of our main result to the stationary qu…
This paper presents a second-order heavy traffic analysis of a single server queue that processes customers having deadlines using the earliest-deadline-first scheduling policy. For such systems, referred to as real-time queueing systems, performance is measured by the fraction of customers who meet their deadline, rather than more traditional performance measures, such as customer delay, queue length or server utilization. To model such systems, one must keep track of customer lead times (the time remaining until a customer deadline elapses) or equivalent information. This paper reviews the e…
We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicat…
In this paper, we analyze a retrial queueing system with Batch Markovian Arrival Processes and two types of customers. The rate of individual repeated attempts from the orbit is modulated according to a Markov Modulated Poisson Process. Using the theory of multi-dimensional asymptotically quasi-Toeplitz Markov chain, we obtain the stability condition and the algorithm for calculating the stationary state distribution of the system. Main performance measures are presented. Furthermore, we investigate some optimization problems. The algorithm for determining the optimal number of guard servers a…
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…
Consider a population of customers each of which needs to decide independently when to arrive to a facility that provides a service during a fixed period of time, say a day. This is a common scenario in many service systems such as a bank, lunch at a cafeteria, music concert, flight check-in and many others. High demand for service at a specific time leads to congestion that comes at a cost, e.g., for waiting, earliness or tardiness. Queueing Theory provides tools for the analysis of the waiting times and associated costs. If customers have the option of deciding when to join the queue, they w…
We study a queueing network with a single shared server, that serves the queues in a cyclic order according to the gated service discipline. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems and robotics. Special cases of the introduced network include well-known polling models and tandem queues. We derive exact limits of the mean delays under both …
We analyze an interacting queueing network on $\mathbb{Z}^d$ that was introduced in Sankararaman-Baccelli-Foss (2019) as a model for wireless networks. We show that the marginals of the minimal stationary distribution have exponential tails. This is used to furnish asymptotics for the maximum steady state queue length in growing boxes around the origin. We also establish a decay of correlations which shows that the minimal stationary distribution is strongly mixing, and hence, ergodic with respect to translations on $\mathbb{Z}^d$.
We consider a queueing system with $n$ parallel queues operating according to the so-called "supermarket model" in which arriving customers join the shortest of $d$ randomly selected queues. Assuming rate $nλ_{n}$ Poisson arrivals and rate $1$ exponentially distributed service times, we consider this model in the heavy traffic regime, described by $λ_{n}\uparrow 1$ as $n\to\infty$. We give a simple expectation argument establishing that majority of queues have steady state length at least $\log_d(1-λ_{n})^{-1} - O(1)$ with probability approaching one as $n\rightarrow\infty$, implying the same …
In this paper, we present a numerical framework for constructing bounds on stationary performance measures of random walks in the positive orthant using the Markov reward approach. These bounds are established in terms of stationary performance measures of a perturbed random walk whose stationary distribution is known explicitly. We consider random walks in an arbitrary number of dimensions and with a transition probability structure that is defined on an arbitrary partition of the positive orthant. Within each component of this partition the transition probabilities are homogeneous. This enab…
In this paper we analyze an $M/M/1$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the…
We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The tool is the detailed balance equations.
We consider a single-server GI/GI/1 queueing system with feedback. We assume the service times distribution to be (intermediate) regularly varying. We find the tail asymptotics for a customer's sojourn time in two regimes: the customer arrives in an empty system, and the customer arrives in the system in the stationary regime. In particular, in the case of Poisson input we use the branching processes structure and provide more precise formulae. As auxiliary results, we find the tail asymptotics for the busy period distribution in a single-server queue with an intermediate varying service times…
We introduce a queueing system that alternates between two modes, so-called {\it working mode} and {\it vacation mode}. During the working mode the system runs as an $M^{X}/G/1$ queue. Once the number of customers in the working mode drops to zero the vacation mode begins. %Then working system becomes empty the vacation phase begins. During the vacation mode the system runs as a general queueing system (a service might be included) which is different from the one in the working mode. The vacation period ends in accordance with a given stopping rule, and then a random number of customers are tr…
A special customer must complete service from two servers in series, in either order, each with an M/M/1 queueing system. It is assumed that the two queueing system lengths are independent with initial numbers of customers a and b at the instant when the special customer arrives. We find the expected total time (ETT) for the special customer to complete service. We show that even if the interarrival and service time parameters of two queues are identical, there exist examples (specific values of the parameters and initial lengths) for which the special customer surprisingly has a lower expecte…
In this paper, we provide a review on the kernel method, which is one of the options for characterizing so-called exact tail asymptotic properties in stationary probabilities of two-dimensional random walks, discrete or continuous (or mixed), in the quarter plane. Many two-dimensional queueing systems can be modelled via these types of random walks. Stationary probabilities are one of the most sought statistical quantities in queueing analysis. However, explicit expressions are available only for a very limited number of models. Therefore, tail asymptotic properties become more important, sinc…
In complex systems, it is quite common to resort to approximations when optimizing system performance. These approximations typically involve selecting a particular system parameter and then studying the performance of the system as this parameter grows without bound. In such an asymptotic regime, we prove that if the approximation to the objective function is accurate up to $\mathcal{O}(1)$, then under some regularity conditions, the prescriptions that are derived from this approximation are $o(1)$-optimal, i.e., their optimality gap is asymptotically zero. A consequence of this result is tha…
