text: 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.
  @cite_35: Many problems arising from the coupling of processors require the solution of functional equations. Generally, the unknown functions are the generating functions for a stationary distribution of the studied process. In this paper, a particular problem is addressed but results lead to a computationally reasonable solution which applies to very general two dimensional random walks.
  @cite_38: In this paper, we consider a two-user and a three-user slotted ALOHA network with multi-packet reception (MPR) capabilities and a queue-aware transmission control. In this setting, 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. We focus on the fundamental problem of characterizing the stable throughput region, as well as of investigating the queueing delay. For the two- and the three-user cases, we obtain the exact stability region, whereas in the former case, we also provide the conditions under which the stability region is a convex set. We perform a detailed mathematical analysis to study the queueing delay in the two-user case by formulating two boundary value problems, 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 the need of solving a boundary value problem. In addition, we provide a closed form expression for the gap between the lower and the upper bound. 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. We also provide a closed form expression for the optimal transmission probability that minimizes the average delay in the symmetric capture case. Finally, we evaluate numerically the presented theoretical results.
  @cite_14: Abstract 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.
  @cite_4: We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann-Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue).
  @cite_33: In this work, we consider an adaptive two-user random access scheme with multipacket reception (MPR) capabilities. A user adapts its transmission characteristics based on the status of the other user. The users have external random traffic stored in their queues. We derive the stable throughput region of the system and the convexity conditions of this region. We derive analytically the queueing delay by formulating and solving a Riemann- Hilbert boundary value problem. Finally, we evaluate numerically the presented theoretical results.
  @cite_7: Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i = 1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann- Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts.
  @cite_36: 
  @cite_9: 
  @cite_3: In this work, we consider an asymmetric two-user random access wireless network with interacting nodes, time-varying links and multipacket reception capabilities. The users are equipped with infinite capacity buffers where they store arriving packets that will be transmitted to a destination node. Moreover, each user employs a general transmission control protocol under which, it adapts its transmission probability based both on the state of the other user, and on the channel state information according to a Gilbert-Elliot model. We study a two-dimensional discrete time Markov chain, investigate its stability condition, and show that its steady state performance is expressed in terms of a solution of a Riemann-Hilbert boundary value problem. Moreover, for the symmetrical system, we provide closed form expressions for the average delay at each user node. Numerical results are obtained and show insights in the system performance.
  @cite_27: We consider a tandem queue with coupled processors and analyze the two-dimensional Markov process representing the numbers of jobs in the two stations. A functional equation for the generating function of the stationary distribution of this two-dimensional process is derived and solved through the theory of Riemann-Hilbert boundary value problems.
  @cite_5: Recently complex function techniques have been developed for the analysis of queueing systems which need for their modelling a two dimensional state space. A variety of computer- and communication networks gives rise to such two-dimensional queueing systems and their analysis is needed for the performance evaluation of these aggregates. The present study reviews these developments
  @cite_15: In Resing and Ormeci [16] it is shown that the two-stage tandem queue with coupled processors can be solved using the theory of boundary value problems. In this paper we consider the issues that arise when calculating performance measures like the mean queue length and the fraction of time a station is empty. It is assumed that jobs arrive at the first station according to a Poisson process and require service at both stations before leaving the system. The amount of work that a job requires at each of the stations is an independent, exponentially distributed random variable. When both stations are nonempty, 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 nonempty station. We study the two-dimensional Markov process representing the numbers of jobs at the two stations. The problem of finding the generating function of the stationary distribution can be reduced to two different Riemann-Hilbert boundary value problems, where both problems yield a complete analytical solution. We discuss the similarities and differences between the two problems, and relate them to the computational aspects of obtaining performance measures.
  @cite_16: Abstract In this work we analyze a novel queueing system for modeling cooperative wireless networks. We consider a network of three saturated source users, say a central and two background users, two relay nodes and a common destination node. When the central user fails to transmit a packet directly to the destination, it forwards a copy of the blocked packet at both relay nodes in order to exploit the spatial diversity they provide. Moreover, each relay node receives also blocked packets from a dedicated background user. Relay nodes assist source users by retransmitting their blocked packets to the destination. Due to the complex interdependence among relays, and the wireless interference, the retransmission rate of a relay is affected by the state of the other. We consider a three-dimensional Markov process, investigate its stability, and study its steady-state performance using the theory of boundary value problems. Explicit expressions for the expected delay in the symmetrical model, and a generalization to N > 2 relay nodes are also given. Numerical examples are obtained and show insights into the system performance.
summary: Queues with coupled processors were initially studied in @cite_35 . To gain quantitative insights about the queueing process, they derived a solution for the generating function of the stationary joint queue-length distribution using the theory of Riemann-Hilbert boundary value problems. Later, in @cite_5 a systematic and detailed study of the technique of reducing a two dimensional functional equation of a random walk or queueing model to a boundary value problem was presented, while several numerical issues were discussed. Important generalizations were given in @cite_9 @cite_7 @cite_4 @cite_15 @cite_27 @cite_36 @cite_16 @cite_3 @cite_33 @cite_38 @cite_14 (not exhaustive list) where various two-dimensional queueing models with the aid of the theory of Riemann (-Hilbert) boundary value problems.