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In this work, we study Markov-dependent reflected autoregressive processes. In queueing terms, such processes describe the workload just before a customer arrival, which makes obsolete a fraction of the work already present, and where the interarrival time and the service time depend on a common discrete time Markov chain. Among others, we consider the case where given the state of the underlying Markov chain, the interarrival time and the service time are independent, as well as the case where there is also additional dependence based on the Farlie-Gumbel-Morgenstern copula, the case where the service time depends on the waiting time of the customer, and the modulated shot-noise queue where server's speed is workload proportional. The Laplace-Stieltjes transform of the steady-state workload is derived via a recursive approach. Moreover, we further consider a Markov-modulated version of the reflected autoregressive process with more general structure and obtain the Laplace-Stieltjes transform of the transient distribution of the workload.

In this work, we consider extensions of the dual risk model with proportional gains by introducing a dependence structure between gain sizes and gain interrarrival times. Among others, we further consider the case where the proportional parameter is randomly chosen, the case where it is a uniformly random variable, as well as the case where we may have upwards as well as downwards jumps. Moreover, we consider the case with causal dependence structure, as well as the case where the dependence is based on the generalized Farlie-Gumbel-Morgenstern copula. The ruin probability and the distribution of the time to ruin are investigated.

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Amongst others, we study cases where the interarrival and service times are proportionally dependent with additive and/or subtracting delay, as well as cases cases where interarrival times depends on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable, as well as cases where . More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted, and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time of a customer in the system in terms of an infinite product of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with orbital searching time to depend on whether the last busy period starts with an empty or a non empty orbit queue, is also discussed. For such a model the probability generating function of the stationary orbit queue length is given as an infinite product of known generating functions. A first attempt towards multidimensional setting is also analyzed. Some additional generalizations with more general dependence structure are also discussed.

Motivated by queueing applications, we consider a certain class of two-dimensional random walks for which their invariant measure is written as a linear combination of a finite number of product-form terms. In this work, we investigate under which conditions such an elegant solution can be derived by applying a finite compensation procedure. The conditions are formulated in terms of relations among the transition probabilities in the inner area, the boundaries as well as the origin. A thorough discussion on the importance of these conditions is also given.

We introduce a novel single-server queue with general retrial times and event-dependent arrivals. This is a versatile model for the study of service systems, in which the server needs a non-negligible time to retrieve waiting customers upon a service completion, while future arrivals depend on the last realized event. Such a model is motivated by the customers' behaviour in service systems where they decide to join based on the last realized event. We investigate the necessary and sufficient stability condition and derive the stationary distribution both at service completion epochs, and at an arbitrary epoch using the supplementary variable technique. We also study the asymptotic behaviour under high rate of retrials. Performance measures are explicitly derived and extensive numerical examples are performed to investigate the impact of event-dependency. Moreover, constrained optimisation problems are formulated and solved with ultimate goal to investigate the admission control problem.

We introduce the \textit{generalized join the shortest queue model with retrials} and two infinite capacity orbit queues. Three independent Poisson streams of jobs, namely a \textit{smart}, and two \textit{dedicated} streams, flow into a single server system, which can hold at most one job. Arriving jobs that find the server occupied are routed to the orbits as follows: Blocked jobs from the \textit{smart} stream are routed to the shortest orbit queue, and in case of a tie, they choose an orbit randomly. Blocked jobs from the \textit{dedicated} streams are routed directly to their orbits. Orbiting jobs retry to connect with the server at different retrial rates, i.e., heterogeneous orbit queues. Applications of such a system are found in the modelling of wireless cooperative networks. We are interested in the asymptotic behaviour of the stationary distribution of this model, provided that the system is stable. More precisely, we investigate the conditions under which the tail asymptotic of the minimum orbit queue length is exactly geometric. Moreover, we apply a heuristic asymptotic approach to obtain approximations of the steady-state joint orbit queue-length distribution. Useful numerical examples are presented, and shown that the results obtained through the asymptotic analysis and the heuristic approach agreed.

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that its stationary behaviour is investigated by solving a finite system of linear equations, two matrix functional equations, and a functional equation with the aid of the theory of Riemann (-Hilbert) boundary value problems. This work is strongly motivated by emerging applications in flow level performance of wireless networks that give rise in queueing models with scalable service capacity, as well as in queue-based random access protocols, where the network's parameters are functions of the queue lengths. A simple numerical illustration, along with some details on the numerical implementation are also presented.

This work introduces the join the shortest queue policy in the retrial setting. We consider a Markovian single server retrial system with two infinite capacity orbits. An arriving job finding the server busy, it is forwarded to the least loaded orbit. Otherwise, it is forwarded to an orbit randomly. Orbiting jobs of either type retry to access the server independently. We investigate the stability condition, the stationary tail decay rate, and obtain the equilibrium distribution by using the compensation method., Comment: 12 pages, 4 figures

In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series approximation method, and the theory of Riemann boundary value problems. We also obtain explicit expressions for the first moments of the stationary distribution under some symmetry assumptions. An application in the modelling of a priority retrial system with coupled orbit queues is also presented. Using a queueing network example, we numerically validated the theoretical findings.

We consider a tandem queue with coupled processors, which is subject to global breakdowns. When the network is in the operating mode and both queues are non empty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the non-empty station. Moreover, arrival rates depend on the state of the network. The system is described by a Markov modulated random walk in the quarter plane representing the number of jobs in the two stations and the state of the network. By applying the generating function approach, we first apply the power series approximation method to obtain power series expansions of the generating function of the stationary queue lengths for both network states. Then, we also provide a way to derive the generating function of the stationary queue lengths for both network states in terms of the solution of a Riemann-Hilbert boundary value problem. Numerical results are obtained to show insights in the system performance.

The effect of signals on stability, throughput region, and delay in a two-user slotted ALOHA based random-access system with collisions is considered. This work gives rise to the development of random access G-networks, which can model virus attacks or other malfunctions and introduce load balancing in highly interacting networks. The users are equipped with infinite capacity buffers accepting external bursty arrivals. We consider both negative and triggering signals. Negative signals delete a packet from a user queue, while triggering signals cause the instantaneous transfer of packets among user queues. We obtain the exact stability region, and show that the stable throughput region is a subset of it. Moreover, we perform a compact mathematical analysis to obtain exact expressions for the queueing delay by solving a Riemann boundary value problem. A computationally efficient way to obtain explicit bounds for the queueing delay is also presented. The theoretical findings are numerically evaluated and insights regarding the system performance are derived.

The scope of this work is twofold: On the one hand, strongly motivated by emerging engineering issues in multiple access communication systems, we investigate the performance of a slotted-time relay-assisted cooperative random access wireless network with collisions and with join the shortest queue relay-routing protocol. For this model, we investigate the stability condition, and apply different methods to derive the joint equilibrium distribution of the queue lengths. On the other hand, using the cooperative communication system as a vehicle for illustration, we investigate and compare three different approaches for this type of multi-dimensional stochastic processes, namely the compensation approach, the power series algorithm (PSA), and the probability generating function (PGF) approach. We present an extensive numerical comparison of the compensation approach and PSA, and discuss which method performs better in terms of accuracy and computation time. We also provide details on how to compute the PGF in terms of a solution of a Riemann-Hilbert boundary value problem., Comment: 33 pages, 4 figures

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks in the quarter plane are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that its stationary behavior is investigated by solving a finite system of linear equations, and a functional equation with the aid of the theory of Riemann(-Hilbert) boundary value problems. This work is strongly motivated by emerging applications in multiple access systems as well as in the study of a general class of queueing systems with state dependent parameters. A simple numerical illustration providing useful information about a queue-aware multiple access system is also presented.

In this work, we consider a random access Internet of Things IoT wireless network assisted by two aggregators collecting information from two disjoint groups of sensors. The nodes and the aggregators are transmitting in a random access manner under slotted time, the aggregators perform network-level cooperation for the data collection. The aggregators are equipped with queues to store data packets that are transmitted by the network nodes and relaying them to the destination node. We characterize the throughput performance of the IoT network and we obtain the stability conditions for the queues at the aggregators. We apply the theory of boundary value problems to analyze the delay performance. Our results show that the presence of the aggregators provides significant gains in the IoT network performance, in addition, we provide useful insights regarding the scalability of the IoT network., Comment: This work was presented in part in the 30th International Teletraffic Congress (ITC). This is the extended version submitted for possible journal publication

We provide power series approximations for a structured batch arrival single server retrial system with two infinite capacity weighted fair orbit queues, i.e., the re-transmission rate of an orbit depends on the state of the other orbit queue. We consider both exponential and arbitrary distributed service times. In both cases we obtain power series expansions of the generating functions of the stationary joint orbit queue-length distributions, and provide a recursive approach to calculate their coefficients. We also show how to obtain the generating function of the stationary joint orbit queue-length distribution with the aid of a Riemann boundary value problem. Power series approximations are also provided for the model with two independent Poisson streams of jobs with single arrivals. Numerical illustrations are performed and show the accuracy of our approach., Comment: submitted paper

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 derived explicit expressions for the expected delay in an orbit without solving a boundary value problem. A simple numerical example is obtained to illustrate the system's performance.

In this work, we investigate a slotted-time relay assisted cooperative random access wireless network with multipacket (MPR) reception capabilities. MPR refers to the capability of a wireless node to successfully receive packets from more than two other modes that transmit simultaneously at the same slot. We consider a network of $N$ saturated sources that transmit packets to a common destination node with the cooperation of two infinite capacity relay nodes. The relays assist the sources by forwarding the packets that failed to reach the destination. Moreover, the relays have also packets of their own to transmit to the destination. We further assume that the relays employ a state-dependent retransmission control mechanism. In particular, a relay node accordingly adapts its transmission probability based on the status of the other relay. Such a protocol is towards self-aware networks and leads to substantial performance gains in terms of delay. We investigate the stability region and the throughput performance for the full MPR model. Moreover, for the asymmetric two-sources, two-relay case we derive the generating function of the stationary joint queue-length distribution with the aid of the theory of boundary value problems. For the symmetric case, we obtain explicit expressions for the average queueing delay in a relay node without solving a boundary value problem. Extensive numerical examples are presented and provide insights on the system performance.

In this work we consider a two-user and a three-user slotted ALOHA network with multi-packet reception (MPR) capabilities. The nodes can adapt their transmission probabilities and their transmission parameters based on the status of the other nodes. Each user has external bursty arrivals that are stored in their infinite capacity queues. For the two- and the three-user cases we obtain the stability region of the system. For the two-user case we provide the conditions where the stability region is a convex set. We perform a detailed mathematical analysis in order to study the queueing delay by formulating two boundary value problems (a Dirichlet and a Riemann-Hilbert boundary value problem), the solution of which provides the generating function of the joint stationary probability distribution of the queue size at user nodes. Furthermore, for the two-user symmetric case with MPR we obtain a lower and an upper bound for the average delay without explicitly computing the generating function for the stationary joint queue length distribution. The bounds as it is seen in the numerical results appear to be tight. Explicit expressions for the average delay are obtained for the symmetrical model with capture effect which is a subclass of MPR models. We also provide the optimal transmission probability in closed form expression that minimizes the average delay in the symmetric capture case. Finally, we evaluate numerically the presented theoretical results., Comment: Submitted for journal publication

In this paper, we investigate the effect of bursty traffic and random availability of caching helpers in a wireless caching system. More explicitly, we consider a general system consisting of a caching helper with its dedicated user in proximity and another non-dedicated user requesting for content. Both the non-dedicated user and the helper have limited storage capabilities. When the user is not able to locate the requested content in its own cache, then its request shall be served either by the caching helper or by a large data center. Assuming bursty request arrivals at the caching helper from its dedicated destination, its availability to serve other users is affected by the request rate, which will further affect the system throughput and the delay experienced by the non-dedicated user. We characterize the maximum weighted throughput and the average delay per packet of the considered system, taking into account the request arrival rate of the caching helper, the request probability of the user and the availability of the data center. Our results provide fundamental insights in the throughput and delay behavior of such systems, which are essential for further investigation in larger topologies.