Phase-type Renewal Process Research Articles

Analyze periodic inspection and replacement policy of a shock and wear model with phase-type inter-shock arrival times using roots method

We consider a shock and wear model in which the inter-shock arrival process is a phase-type (PH) renewal process, and the system’s lifetime is generally distributed. The system has two competing failure modes. One failure mode is due to random shocks, which cause failure by overloading the system. The other failure mode is owning to wear-out failures, which usually happen after the system has run for many cycles. System failure is not self-announcing and remains undiscovered unless an inspection is performed. The intervals between successive inspections are identical and equal to T time units. If a system failure is detected, the corrective repair or replacement is conducted immediately. If the system is found working at inspection, preventive maintenance will be carried out to prolong its useful life. Furthermore, to model the occurrence of events with an underlying monotonic trend, the extended geometric process (EGP) is employed to account for the impact of different types of failures on the system’s degradation. Moreover, for establishing the cost rate function in our model, the counting process generated from a PH renewal shock process is studied in detail using the roots method and formula for calculating residues. Based on these results, the survival function and other characteristics of the system are further investigated. Finally, numerical examples that determine the optimal inspection period T[Formula: see text] and the optimal replacement policy N[Formula: see text], which minimizes the long-run average cost rate, are presented.

Reliability Evaluation and Optimization of a System with Mixed Run Shock

In this paper, we investigate a wear and mixed shock model in which the system can fail due to internal aging or external shocks. The lifetime of the system, due to internal wear, follows continuous phase-type (PH) distributions. The external random shocks arrive at the system according to a PH renewal process. The system will fail when the internal failure occurs or k1 consecutive external shocks, the size of at least d1 or k2 consecutive external shocks the size of at least d2 occur, where d1<d2, k1>k2. The failed system can be repaired immediately, and the repair times of the system are governed by continuous PH distributions. The system can be replaced by a new and identical one based on a bivariate replacement policy (L,N). The long-run average profit rate for the system is obtained by employing the closure property of the PH distribution. Finally, a numerical example is also given to determine the optimal replacement policy.

TTL approximations of the cache replacement algorithms LRU(m) and h-LRU

Computer system and network performance can be significantly improved by caching frequently used information. When the cache size is limited, the cache replacement algorithm has an important impact on the effectiveness of caching. In this paper we introduce time-to-live (TTL) approximations to determine the cache hit probability of two classes of cache replacement algorithms: h-LRU and LRU(m). These approximations only require the requests to be generated according to a general Markovian arrival process (MAP). This includes phase-type renewal processes and the IRM model as special cases. We provide both numerical and theoretical support for the claim that the proposed TTL approximations are asymptotically exact. In particular, we show that the transient hit probability converges to the solution of a set of ODEs (under the IRM model), where the fixed point of the set of ODEs corresponds to the TTL approximation.We use this approximation and trace-based simulation to compare the performance of h-LRU and LRU(m). First, we show that they perform alike, while the latter requires less work when a hit/miss occurs. Second, we show that as opposed to LRU, h-LRU and LRU(m) are sensitive to the correlation between consecutive inter-request times. Last, we study cache partitioning. In all tested cases, the hit probability improved by partitioning the cache into different parts—each being dedicated to a particular content provider. However, the gain is limited and the optimal partition sizes are very sensitive to the problem’s parameters.

Energy Efficiency Optimization of Secondary Network Considering Primary User Return With Alternating-Phase-Type Traffic

Much work has been done in the area of cognitive radio networks to formulate the inherent tradeoff among different metrics, such as sensing and throughput. Primary user (PU) traffic behavior can affect the mentioned tradeoff and must be considered in analyses, especially when cognitive users coexist with the dynamic-traffic PU. In this case, PU-returns to the spectrum during secondary users' (SUs) transmission are more likely and, hence, such kinds of interference are inevitable. Furthermore, how to model the PU traffic is important. Note that there are two kinds of interference between SU and PU in cognitive radio networks, one due to sensing error and the other due to primary user reoccupancy. In this paper, a new metric called energy per successful transmission time is defined to address both kinds of inference by formulating SUs' successful transmission time as well as energy consumption. Furthermore, collision probability and average packet delay are formulated considering these interference metrics and the primary user traffic is modeled as an alternating phase type renewal process by which many traffic behaviors, such as long range dependency and self-similarity can be modeled. Finally, some numerical examples are given and the corresponding curves for different metrics are discussed.

On the Queue Length Distribution in BMAP Systems

Batch Markovian Arrival Process – BMAP – is a teletraffic model which combines high ability to imitate complexstatistical behaviour of network traces with relative simplicity in analysis and simulation. It is also a generalization of a wide class of Markovian processes, a class which in particular include the Poisson process, the compound Poisson process, the Markovmodulated Poisson process, the phase-type renewal process and others. In this paper we study the main queueing performance characteristic of a finite-buffer queue fed by the BMAP, namely the queue length distribution. In particular, we show a formula for the Laplace transform of the queue length distribution. The main benefit of this formula is that it may be used to obtain both transient and stationary characteristics. To demonstrate this, several numerical results are presented.

The Distribution of Discounted Compound PH–Renewal Processes

We continue and extend the work in Leveille et al. (2010) Scand Actuar J 3:165–184 that gives analytical formulas for the moment generating function (mgf) of some discounted compound renewal processes. Here these mgf’s are used to derive and study the distribution of discounted compound Phase-type (PH) renewal sums. The approach consists in first deriving a differential equation, in the time variable, for the mgf of discounted compound sums when the inter–arrival times are PH–distributed. Then the corresponding distribution of the discounted compound PH sum is obtained by inversion of its mgf (or the equivalent Laplace transform). Analytical expressions for the asymptotic distribution of these discounted compound renewal sums are also given, to test general expressions in some limiting cases. Despite the technical difficulty, several new examples are provided where the inversion is possible, through symbolic computation in MAPLE. When the inversion is too complex, a truncated series solution method is proposed, which is not new but turns out to be very fast and accurate here, much more than other methods (such as rational approximations). The article concludes with some applications where the mgf and the distribution are used to calculate risk measures such as VaR, CTE, Esscher’s, Wang’s Proportional Hazard transforms and evaluate their evolution in time.

On the Relation between Phase-Type Distributions and Positive Systems

The relation between phase-type representation and positive system realization in both the discrete and continuous time is discussed. Using the Perron-Frobenius theorem of nonnegative matrix theory, a transformation from positive realization to phase-type realization is derived under the excitability condition. In order to explain the connection, some useful properties and characteristics such as irreducibility, excitability, transparency, and order reduction for positive realization and phase-type representation are discussed. In addition, the connection between the phase-type renewal process and the feedback positive system is discussed in the stabilization concept.

Analysis of a multi-server queueing model with MAP arrivals of regular customers and phase type arrivals of special customers

We consider a multi-server queueing model in which two types of arrivals occur. Regular customers arrive according to a Markovian arrival process and special customers arrive according to a phase type renewal process. The service times are exponentially distributed with parameter depending on the type served. While a regular customer requires the attention of only one server, a special customer needs the attention of all servers. An arriving special customer finding the servers busy with a special customer is lost; otherwise, the special customer will enter into service immediately by possibly pre-empting existing services of regular customers. The displaced customers will enter into service once the special customer leaves. The model is analyzed in steady state and some illustrative numerical examples are discussed to bring out the key effect of the special customers on selected system performance measures.

Optimal order-replacement policy for a phase-type geometric process model with extreme shocks

A system is subject to random shocks that arrive according to a phase-type (PH) renewal process. As soon as an individual shock exceeds some given level the system will break down. The failed system can be repaired immediately. With the increasing number of repairs, the maximum shock level that the system can withstand will be decreasing, while the consecutive repair times after failure will become longer and longer. Undergoing a specified number of repairs, the existing system will be replaced by a new and identical one. The spare system for the replacement is available only by sending a purchase order to a supplier, and the duration of spare system procurement lead time also follows a PH distribution. Based on the number of system failures, a new order-replacement policy (also called (K,N) policy) is proposed in this paper. Using the closure property of the PH distribution, the long-run average cost rate for the system is given by the renewal reward theorem. Finally, through numerical calculation, it is determined an optimal order-replacement policy such that the long-run expected cost rate is minimum.

Numerical investigation of a PH/PH/1 inventory system with positive service time and shortage

The aim of this paper is to numerically investigate a PH/PH/1 inventory model with reneging of customers and finite shortage of items. We assume that arrivals occur according to a phase type renewal process. The associated phase type distribution has representation (α, U). The service times are identically and independently distributed random variables having common phase type distribution with representation (β, V). The lead-time is zero. Costumers renege from the system at a constant rate γ. Shortage is permitted and hence shortage cost is finite. We perform the steady state analysis of the inventory model using Matrix analytic method. A suitable cost function is defined and analyzed numerically. The optimal shortage level is numerically evaluated. Some measures of the system performance in the steady state are also derived.

A maintenance model with failures and inspection following Markovian arrival processes and two repair modes

We consider a deteriorating system submitted to external and internal failures, whose deterioration level is known by means of inspections. There are two types of repairs: minimal and perfect, depending on the deterioration level, each one following a different phase-type distribution. The failures and the inspections follow different Markovian arrival processes (MAP). Under these assumptions, the system is governed by a generalized Markov process, whose state space and generator are constructed. This general model includes the phase-type renewal process as a special case. The distribution of the number of minimal and perfect repairs between two inspections are determined. A numerical application optimizing costs is performed, and different particular cases of the model are compared.

A Discrete-Time Model for Common Lifetime Inventory Systems

We consider a discrete-time (s, S) inventory model in which the stored items have a random common lifetime with a discrete phase-type distribution. Demands arrive in batches following a discrete phase-type renewal process. With zero lead time and allowing backlogs, we construct a multidimensional Markov chain to model the inventory-level process. We obtain a closed-form expected cost function. Numerical results demonstrate some properties of optimal ordering policies and cost functions. When compared with the results for the constant lifetime model, the variance of the lifetime significantly affects the system behavior. Thus, the formalism that we create here adds a new dimension to the research in perishable inventory control under uncertainty in lifetime.

Approximation method for M/PH/1 retrial queues with phase type inter-retrial times

We obtain a stability condition and develop a method for approximating the stationary distribution and waiting time moments of an M/PH/1 retrial queue with phase type inter-retrial times. This extends a previous method applicable for general service and retrial times which approximates only the stationary queue length distribution. Our approximation replaces the complex non-renewal process of secondary arrivals from the orbit with a phase type renewal process which matches the first two or three interarrival time moments. The queue is modelled as a quasi birth–death process with subdiagonal blocks of rank one. The stationary distribution is obtained by applying a known result for generators of this form and moments of the waiting time are obtained numerically by applying block Gaussian elimination to a truncated system. The performance of the approximation is evaluated on a set of instances for which exact solutions are obtainable. We also describe how the method can be used to approximate the distribution of the waiting time and the number of retrials.

Further transient analysis of the BMAP/G/1 Queue

Previously, we derived the two-dimensional transforms of the emptiness function, the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, as well as superpositions of these processes, and allows correlated interarrival times and batch sizes We continue the transient analysis of this model in this paper by deriving explicit expressions for the transforms of the queue length at the n-th departure (assuming a departure at time t = 0), and the delay of the n-th arrival (keeping track of the appropriate phase changes). Also, the departure process is characterized by the double transform of the probability that the n -th departure occurs at time less than or equal to time x

Light traffic approximations for regenerative queueing processes

We consider a regenerative queueing process that is (partially) generated by an embedded phase-type renewal process. We show that, under some specified conditions, a performance measure is an analytic function of the rate of the renewal process. We then develop several methods for deriving its Taylor polynomial in the renewal rate. These polynomials are asymptotically exact as the rate decreases, and, thus, are called light traffic approximations of the performance measure. We show via examples that these new methods are not only more efficient compared to existing ones, but also more versatile due to their general settings, such as to conduct perturbation analysis and study transient behavior.

Light traffic approximations for regenerative queueing processes

We consider a regenerative queueing process that is (partially) generated by an embedded phase-type renewal process. We show that, under some specified conditions, a performance measure is an analytic function of the rate of the renewal process. We then develop several methods for deriving its Taylor polynomial in the renewal rate. These polynomials are asymptotically exact as the rate decreases, and, thus, are called light traffic approximations of the performance measure. We show via examples that these new methods are not only more efficient compared to existing ones, but also more versatile due to their general settings, such as to conduct perturbation analysis and study transient behavior.

A numerically efficient method for the MAP/D/1/K queue via rational approximations

The Markovian Arrival Process (MAP), which contains the Markov Modulated Poisson Process (MMPP) and the Phase-Type (PH) renewal processes as special cases, is a convenient traffic model for use in the performance analysis of Asynchronous Transfer Mode (ATM) networks. In ATM networks, packets are of fixed length and the buffering memory in switching nodes is limited to a finite numberK of cells. These motivate us to study the MAP/D/1/K queue. We present an algorithm to compute the stationary virtual waiting time distribution for the MAP/D/1/K queue via rational approximations for the deterministic service time distribution in transform domain. These approximations include the well-known Erlang distributions and the Padé approximations that we propose. Using these approximations, the solution for the queueing system is shown to reduce to the solution of a linear differential equation with suitable boundary conditions. The proposed algorithm has a computational complexity independent of the queue storage capacityK. We show through numerical examples that, the idea of using Padé approximations for the MAP/D/1/K queue can yield very high accuracy with tractable computational load even in the case of large queue capacities.

Transient analysis of fluid model for ATM statistical multiplexer

We analyze the transient behavior of a fluid model of ATM multiplexer, in which the input process is generated by a Markov modulated rate process (MMR Process). This model is a natural extension of the on-off-type multi-entry queueing model which has been widely used for modeling ATM multiplexers with bursty inputs. The Laplace transform of the joint distribution of the buffer content and the state of the input process is obtained by solving a system of partial differential equations. The unknown functions included in the solution are determined from the boundary conditions by using eigenvalues and eigenvectors of a key matrix. By taking a limit of the solution, the state probabilities in the steady-state are written explicitly. An effective lower bound of the relaxation time is also presented. In case of an on-off-type input fluid model with phase-type on/off period distributions, the equations for the eigenvalues and eigenvectors of the key matrix are reduced to more concrete ones using the Laplace transforms of the on/off period distributions. The lower bound of the relaxation time is also reduced to the maximum among the relaxation times of phase-type renewal processes governing the on/off periods of the inputs.

Some performance measures for vacation models with a batch Markovian arrival process

We consider a single server infinite capacity queueing system, where the arrival process is a batch Markovian arrival process (BMAP). Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP includes phase‐type (PH) renewal processes and non‐renewal processes such as the Markov modulated Poisson process (MMPP).The server applies Kella′s vacation scheme, i.e., a vacation policy where the decision of whether to take a new vacation or not, when the system is empty, depends on the number of vacations already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T‐policy, T(SV), and the multiple vacation T‐policy, T(MV). The service times are i.i.d. random variables, independent of the interarrival times and the vacation durations. Some important performance measures such as the distribution functions and means of the virtual and the actual waiting times are given. Finally, a numerical example is presented.

The transient BMAP/G/l queue

We derive the two-dimensional transforms of the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, as well as superpositions of these processes, and allows correlated interarrival times and batch sizes. Numerical results are obtained via two-dimensional transform inversion algorithms based on the Fourier-series method. From the numerical examples we see that predictions of system performance based on transient and stationary performance measures can be quite different