Phase type delay distribution with PU return and sensing error in CR networks

Many studies have been done in cognitive radios to maximize the network efficiency. There is always a trade-off between sensing and transmission time which has been addressed fully in the literature. There is also some missed idle times during the waiting phase when secondary user finds the channel busy. Therefore, the longer the transmission time is, the higher the successfully delivered bits would be and the higher the missed idle times in the waiting stage would be expected. In this work, we formulate these missed idle times to addressed this trade-off. Furthermore, interference due to PU re-occupancy is modelled in successful transmitted time and in collision probability. Moreover, we derive secondary user's packet delay distribution using phase type model at which retransmission due to collision results from both sensing error and PU re-occupancy is considered. Thanks to derived delay distribution, any statistical moments of delay could be found as a closed form mathematical expression. These moments can be used as constraints for applications with delay sensitive packets. Numerical results are given to clarify the network metrics. Finally, the optimized values for sensing and transmission time are computed using genetic algorithm in order to maximize access efficiency.

A Maintenance Model Based on Phase Type Distribution for Equipment under Two Types of Failures

A maintenance model is put forward for equipment, which is subjected to internal and external failures. In the model, two maintenance pollcies--perfect minimal repair and replacement are considered. The operational time has a phase type distribution （PH distribution）. Failures＇ arrival follows a Markovian Arrival Process （MAP）. Some failures require the replacement of the system, and others minimal repairs. The performance measures-the mean number of minimal repairs and replacements are analyzed and calculated.

The MAP/PH(PH/PH)/1 Discrete-time Queuing System with Repairable Server

In this paper, we discuss a discrete time repairable queuing system with Markovian arrival process, where lifetime of server, service time and repair time of server are all discrete phase type random variables. Using the theory of matrix geometric solution, we give the steady state distribution of queue length and waiting time. In addition, the stable availability of the system is also provided.

ON THE SPH-DISTRIBUTION CLASS

Following up Neuts' idea, the SPH-distribution class associated with bounded Q matrices for infinite Markov chains is denned. The main result in this paper is to characterize the SPH class through the derivatives of the distribution functions. Based on the characterization theorem, closure properties, the expansion, uniform approximation, and the matrix representations of the SPH class are also discussed by the derivatives of the distribution functions at origin.

New bounds of (s,S) policies in periodical review inventory systems

We study the stochastic inventory problem with optimal （s,S） policies.In a finite horizon model with lost sales,we establish new lower and upper bounds of s and S.These bounds have structural implications for the optimal solutions.Consequently,when demand has a generalized phase type distribution,there are no more than a pre-determined number of minima.Similar bounds can also be found for the system where unsatisfied demand is backordered instead of lost sales.

Reliability Study on the Risk Precontrol Management System of Coal Mines Safety

Risk precontrol management system of coal mines safety( RPMSCS) provides a set of preventive safety management strategy for high-risk coal industries, which has captured extensive attentions. Fundamentally,there are several membership systems with subsystems in the management system, and the subsystem reliability has an important influence on the management system performance. Through analyzing the structure characteristics of the management system,the phase type distribution was employed to analyze its subsystem reliability by considering repair process and three states including working,fail-abnormal,and fail-emergency states. The reliability indices of the subsystem were derived respectively,including the probabilities that the subsystem in three states,mean time to the first failure, mean time to first failemergency,mean working time to first fail-emergency,and mean maintenance time to the first fail-emergency, are derived respectively. The probabilities of the membership systems and the management system in three states were also derived. Some numerical examples were used to show the procedures. The result is important for better understanding the management system operation and improving its operational performance from the respect of system reliability.

On Numerical Approach to Non-Markovian Stochastic Systems Modeling

The paper considers the problem of representing non-Markovian systems that evolve stochastically over time. It is often necessary to use approximations in the case the system is non-Markovian. Phase type distribution is by now indispensable tool in creation of stochastic system models. The paper suggests a method and software for evaluating stochastic systems approximations by Markov chains with continuous time and countable state space. The performance of a system is described in the event language used for generating the set of states and transition matrix between them. The example of a numerical model is presented.

THE M/PH/1 QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

We study an M/PH/1 queue with phase type working vacation and vacation interruption where the vacation time follows a phase type distribution. The server serves the customers at a lower rate in a vacation period. The server comes back to the regular busy period at a service completion without completing the vacation. Such policy is called vacation interruption. In terms of quasi birth and death process and matrix-geometric solution method, we obtain the stationary queue length distribution. Moreover we obtain the conditional stochastic decomposition structures of queue length and waiting time when the service time distribution in the regular busy period is exponential.