$M|G|\infty$ Queue Parameters Values Approximation Through a Markov Renewal Process

6 Dependence notions in reliability theory

We define a notion of regularity ordering among stochastic processes called directionally convex (dcx) ordering and give examples of doubly stochastic Poisson and Markov renewal processes where such ordering is prevalent. Further-more, we show that the class of segmented processes introduced by Chang, Chao, and Pinedo [3] provides a rich set of stochastic processes where the dcx ordering can be commonly encountered. When the input processes to a large class of queueing systems (single stage as well as networks) are dcx ordered, so are the processes associated with these queueing systems. For example, if the input processes to two tandem /M/c1→/M/c2→…→/M/cm queueing systems are dcx ordered, so are the numbers of customers in the systems. The concept of directionally convex functions (Shaked and Shanthikumar [15]) and the notion of multivariate stochastic convexity (Chang, Chao, Pinedo, and Shanthikumar [4]) are employed in our analysis.

This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

This paper presents an extension to the the Ivanoff–Merzbach renewal process. The Ivanoff–Merzbach process is a natural analogue to the one-dimensional renewal process. Restricting our attention to two dimensions, the focus of the current paper will be formation of this extension, which we will refer to as a Markov renewal process, as well as simulation, estimation and applications to data sets in ecology and forestry. Copyright © 2010 John Wiley & Sons, Ltd.

Departure process of a single server queueing system with Markov renewal input and general service time distribution

At first there was the Markov property. The theory of stochastic processes, which can be considered as an exten- sion of probability theory, allows the modeling of the evolution of systems through the time. It cannot be properly understood just as pure mathemat- ics, separated from the body of experience and examples that have brought it to life. The theory of stochastic processes entered a period of intensive develop- ment, which is not finished yet, when the idea of the Markov property was brought in. Not even a serious study of the renewal processes is possible without using the strong tool of Markov processes. The modern theory of Markov processes has its origins in the studies by A. A: Markov (1856-1922) of sequences of experiments connected in a chain and in the attempts to describe mathematically the physical phenomenon known as Brownian mo- tion. Later, many generalizations (in fact all kinds of weakenings of the Markov property) of Markov type stochastic processes were proposed. Some of them have led to new classes of stochastic processes and useful applications. Let us mention some of them: systems with complete connections [90, 91, 45, 86]; K-dependent Markov processes [44]; semi-Markov processes, and so forth. The semi-Markov processes generalize the renewal processes as well as the Markov jump processes and have numerous applications, especially in relia- bility.

We derive explicit closed expressions for the moment generating functions of whole collections of quantities associated with the waiting time till the occurrence of composite events in either discrete or continuous-time models. The discrete-time models are independent, or Markov-dependent, binary trials and the events of interest are collections of successes with the property that each two consecutive successes are separated by no more than a fixed number of failures. The continuous-time models are renewal processes and the relevant events are clusters of points. We provide a unifying technology for treating both the discrete and continuous-time cases. This is based on first embedding the problems into similar ones for suitably selected Markov chains or Markov renewal processes, and second, applying tools from the exponential family technology.

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

Aven and Jensen (1999) proposed an approximation to the downtime distribution for arbitrary coherent systems with binary and independent repairable components, where each component follows an alternating renewal process. The approximation is based on a mixture of the duration distributions of the cut sets in the system. We extend this approach to coherent systems composed by multistate independent components, where every component follows a semi-Markov process and where we introduce the concept of a minimal cut set in the multistate setting. We test our proposal on a simple power demand-generation system, by comparing the analytic approximation with simulation results, and we find it is accurate when the system is highly available.

For complex equipment and systems, reliability analysis is generally performed at two differents levels. At subassembly level, the designer performs failure rate and failure mode analyses to check fulfilment of reliability requirements, and to detect and eliminate reliability weaknesses as early as possible in the design phase. At equipment and system level, the reliability engineer also investigates time behaviour, taking into account reliability, maintainability, and logistical aspects. Depending upon the system complexity, upon the assumed distribution functions for failure-free and repair times, and with thought toward maintenance policy, investigations are performed either analytically, making use of stochastic processes, or numerically with the help of Monte Carlo simulations. Stochastic processes used in the modeling of reliability problems include renewal and alternating renewal processes, Markov processes with a finite state space, semi-Markov processes, regenerative stochastic processes with only one (or a few) regeneration state(s), and some kinds of non-regenerative stochastic processes. The reliability models covered by each of these processes are given in Table 1.KeywordsRegeneration StateRepair TimeReliability FunctionRepair RateMaintenance PolicyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

. This article focuses on the optimal maintenance policy of a system under multi-state with the unit damage area and repair time based on Markov renewal process. In our model, the system performs planned maintenance for a time interval T without failure, and it undergoes detecting the degree of damage immediately when the fail occurs, if the unit damage area is completed within a thresholds for minor repair and update state S, the system is returned to the minor repair state and repaired failed unit by workers, otherwise it returned to the update state and replaced by new one. At first, the mathematical model with the optimal objective is proposed, which is maximizing the expected reward rate. Next, in order to solve this model, we derive some theorems of renewal functions and transition probabilities in a Markov renewal process with multi-state: operation, detection of damage area, planned maintenance, minor repair, and update. Finally, the numerical example shows that the analytical solution of the optimal planned maintenance time T * and critical value of the damage area S * are obtained when the lifetime distribution is given; moreover, the sensitivity analysis is illustrated to validate the effectiveness of the optimal maintenance policy.

In this paper, Markov Renewal processes having a finite number of states are studied. Explicit expressions are derived for the distribution functions of first passage times, as well as for the marginal distribution function of the corresponding Semi-Markov process. Double generating functions are obtained for the distribution functions of the $N_j$-processes. The limiting behavior of a Markov Renewal process is discussed, the stationary probabilities being derived completely. General Markov Renewal processes are introduced, and a related stationary process is determined. Several examples are given.

A Markov Renewal Process (M.R.P.) is a process similar to a Markov chain, except that the time required to move from one state to another is not fixed, but is a random variable whose distribution may depend on the two states between which the transition is made. For an M.R.P. ofm (<∞) states we derive a goodness-of-fit test for a hypothetical matrix of transition probabilities. This test is similar to the test Bartlett has derived for Markov chains. We calculate the first two moments of the test statistic and modify it to fit the moments of a standard χ2. Finally, we illustrate the above procedure numeerically for a particular case of a two-state M.R.P.

Sleep modes are important for user equipment to reduce power consumption or to reduce battery reloading. To keep the user equipment connected, it has to be activated for uplink or downlink data transfer which causes overhead and additional delay. Wireless or mobile equipment has to be monitored repeatedly for location updates. These classes of problems can be modeled by queuing systems with modified busy and idle periods. In this paper, we consider a generalized infinite-buffer, single-server queuing systems with activation overhead and under various types of sleep modes during inactive periods. For the special assumption of Markovian arrival processes of data units and generally distributed activation times and vacation (sleep) periods, the models can be analyzed exactly by a mean value analysis method which is based on regenerative stochastic processes, renewal theory and classical queuing theorems. The method allows also for dynamic sleep periods which are dependent on each other as in the case of the standard IEEE 802.16m for wireless equipment. Explicit results are derived for average values of data unit delays, resource utilization and power- saving margins dependent on given parameters for activation times and wakeup signaling frequencies. The models include the special cases of M/G/1 queuing systems with a modified service time of the first arrival initiating a busy period and M/G/1 queuing systems with vacation periods, respectively, which are well known from literature providing general solutions for the state and delay distributions. Finally, all models are extended to the exact analysis of Batch Poisson arrival processes.

This paper is concerned with an evaluation method in which the arrival process of burst packets is approximated by a hyperexponential distribution (H2), and the performance of a queueing system (ΣH2/G/1) with a superposed flow as the input is approximated by H2/G/1. This approach approximates the superposition process by a renewal process. The QNA method and Albin's method are currently used for this purpose. Since these existing methods approximate the arrival process and its superposition by using the moments up to second order, they cannot be applied to the evaluation of a ΣH2/G/1 system involving superposition of burst packets with arbitrary degrees of distortion (corresponding to the third-order moment). In this paper, the burst packets are approximated by an asymmetric H2 distribution that can take into account moments up to third order. A method is proposed to evaluate the superposition process by approximation with the renewal process of H2. When the superposition of the asymmetric H2 is approximated by H2, the methods of estimation of the second- and third-order moments of the superposition process are important. A method is described for estimation of each moment from the index of dispersion for the count l(t) of the number of arriving burst packets. Computer simulation of the mean waiting time and comparison with QNA demonstrates the effectiveness of the proposed performance evaluation. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 1, 80(7): 77–89, 1997