Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N  -policy GI/Geo/1$\mathit{GI}/\mathit{Geo}/1$ queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.     

The N-policy of a discrete time Geo/G/1 queue with disasters and its application to wireless sensor networks

We analyze an N-policy of a discrete time Geo/G/1 queue with disasters. We obtain the probability generating functions of the queue length, the sojourn time, and regeneration cycles such as the idle period and the busy period. We apply the queue to a power saving scheme in wireless sensor networks under unreliable network connections where data packets are lost by external attacks or shocks. We present various numerical experiments for application to power consumption control in wireless sensor networks. We investigate the characteristics of the optimal N-policy that minimizes power consumption and derive practical insights on the operation of the N-policy in wireless sensor networks.     

The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

Queue length and waiting time of the M/G/1 queue under the D-policy and multiple vacations

We study the steady-state queue length and waiting time of the M/G/1 queue under the D-policy and multiple server vacations. We derive the queue length PGF and the LSTs of the workload and waiting time. Then, the mean performance measures are derived. Finally, a numerical example is presented and the effects of employing the D-policy are discussed. AMS Subject Classifications 60K25 This work was supported by the SRC/ERC program of MOST/KOSEF grant # R11-2000-073-00000.     

An extended replacement policy for a deteriorating system with multi-failure modes

In this paper, the maintenance problem for a deteriorating system with k + 1 failure modes, including an unrepairable failure (catastrophic failure) mode and k repairable failure (non-catastrophic failure) modes, is studied. Assume that the system after repair is not “as good as new” and its deterioration is stochastic. Under these assumptions, an extended replacement policy N is considered: the system will be replaced whenever the number of repairable failures reaches N or the unrepairable failure occurs, whichever occurs first. Our purpose is to determine an optimal extended policy N^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal extended policy N^∗ can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results of the repair model proposed in this paper.     

Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.     

Controlling arrivals for a queueing system with an unreliable server: Newton-Quasi method

This paper deals with the control policy of a removable and unreliable server for an M/M/1/K queueing system, where the removable server operates an F-policy. The so-called F-policy means that when the number of customers in the system reaches its capacity K (i.e. the system becomes full), the system will not accept any incoming customers until the queue length decreases to a certain threshold value F. At that time, the server initiates an exponential startup time with parameter γ and starts allowing customers entering the system. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. A matrix analytical method is applied to derive the steady-state probabilities through which various system performance measures can be obtained. A cost model is constructed to determine the optimal values, say (F^∗, μ^∗, γ^∗), that yield the minimum cost. Finally, we use the two methods, namely, the direct search method and the Newton-Quasi method to find the global minimum (F^∗, μ^∗, γ^∗). Numerical results are also provided under optimal operating conditions.     

Analysis of a two-phase queueing system with a fixed-size batch policy

We consider a single-server, two-phase queueing system with a fixed-size batch policy. Customers arrive at the system according to a Poisson process and receive batch service in the first-phase followed by individual services in the second-phase. The batch service in the first-phase is applied for a fixed number (k) of customers. If the number of customers waiting for the first-phase service is less than k when the server completes individual services, the system stays idle until the queue length reaches k. We derive the steady state distribution for the system’s queue length. We also show that the stochastic decomposition property can be applied to our model. Finally, we illustrate the process of finding the optimal batch size that minimizes the long-run average cost under a linear cost structure.     

The effects of a single stage-structured population model with impulsive toxin input and time delays in a polluted environment

Uncontrolled contribution of pollutant to the environment has led many species to extinction and several others are at the verge of extinction. This article deals with the dynamics of a single stage-structured population model with impulsive toxin input and time delays (including constant individual maturation time delay and pollution time delay) in a polluted environment, in which we assume that only the mature individuals are affected by pollutants. We obtain conditions for the global attractivity of the population-extinction periodic solution and the permanence of the population. We show that maturation time delay and impulsive toxin input can bring great effects on the dynamics of the system, and pollution time delay is harmless. Numerical simulations confirm our theoretical results.     

Analysis of an M/G/∞ queue with batch arrivals and batch-dedicated servers

We analyze an M/G/∞ queue with batch arrivals, where jobs belonging to a batch have to be processed by the same server. The number of jobs in the system is characterized as a compound Poisson random variable through a scaling of the original arrival and batch size processes.     

Operating characteristic analysis on the M/G/1 system with a variant vacation policy and balking

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a variant vacation policy, where the server leaves for a vacation as soon as the system is empty. The server takes at most J vacations repeatedly until at least one customer is found waiting in the queue when the server returns from a vacation. If the server is busy or on vacation, an arriving batch balks (refuses to join) the system with probability 1 − b. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Finally, important system characteristics are derived along with some numerical illustration.     

Waiting time distribution of the MAP/D/k system in discrete time—a more efficient algorithm

We show that the discrete time MAP/D/k presented by Chaudhry et al. (Oper. Res. Lett. 30(3) (2002) 174) has a special structure which results in a simple and more efficient computational scheme than they have presented. Specifically, we show that the computational efforts for the matrix G at each iteration can be reduced from O(d³k³m³) to O(dk³m³) by rearranging the state space and then capitalizing on the resulting structure. This saving in computational effort is significant, especially when d is very large.     

Algorithmic analysis of the Geo/Geo/c retrial queue

In this paper, we consider a discrete-time queue of Geo/Geo/c type with geometric repeated attempts. It is known that its continuous counterpart, namely the M/M/c queue with exponential retrials, is analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused from the retrial feature. In discrete-time, the occurrence of multiple events at each slot increases the complexity of the model and raises further computational difficulties. We propose several algorithmic procedures for the efficient computation of the main performance measures of this system. More specifically, we investigate the stationary distribution of the system state, the busy period and the waiting time. Several numerical examples illustrate the analysis.     

Stabilization of a coupled transport-diffusion system with boundary input

We consider the problem of stabilizing a coupled transport-diffusion system with boundary input. The system is described by two linear transport-diffusion equations and is not asymptotically stable. In order to stabilize the system with boundary input, sensor influence functions are assumed to be located at interior of the domain. First, we formulate the system as an evolution equation with unbounded output operators in a Hilbert space, using variable transformation. Next, we derive a reduced-order model with a finite-dimensional state variable for the infinite-dimensional system. Then, a stabilizing controller is constructed for the reduced-order model under an additional assumption. It is shown that the finite-dimensional controller together with a residual mode filter plays a role of a finite-dimensional stabilizing controller for the original infinite-dimensional system, if the order of the residual mode filter is chosen sufficiently large. Finally, the validity of the design method is demonstrated through a numerical simulation.     

Design of a production system with a feedback buffer

In this paper, we deal with an M/G/1 Bernoulli feedback queue and apply it to the design of a production system. New arrivals enter a “main queue” before processing. Processed items leave the system with probability 1-p or are fed back with probability p into an intermediate finite “feedback queue”. As soon as the feedback queue is fully occupied, the items in the feedback queue are released, all at a time, into the main queue for another processing. Using transform methods, various performance measures are derived such as the joint distribution of the number of items in each queue and the dispatching rate. We then derive the optimal buffer size which minimizes the overall operating cost under a cost structure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Operating characteristics of MX/G/1 queue with N-policy

We consider aM ^X/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theM ^X/G/1 queueing system withoutN-policy and the other one has the probability generating function _j=0 ^N=1 _j z ^j/ _j=0 ^N=1 _j , in which _j is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.     

Algorithm of robust stability region for interval plant with time delay using fractional order PID controller

By using PI^λD^μ controller, we investigate the problem of computing the robust stability region for interval plant with time delay. The fractional order interval quasi-polynomial is decomposed into several vertex characteristic quasi-polynomials by the lower and upper bounds, in which the value set of the characteristic quasi-polynomial for vertex quasi-polynomials in the complex plane is a polygon. The D-decomposition technique is used to characterize the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters. We investigate how the fractional integrator order λ and the derivative order μ in the range (0, 2) affect the stabilizability of each vertex characteristic quasi-polynomial. The stability region of interval characteristic quasi-polynomial is determined by intersecting the stability region of each quasi-polynomial. The parameters of PI^λD^μ controller are obtained by selecting the control parameters from the stability region. Using the value set together with zero exclusion principle, the robust stability is tested and the algorithm of robust stability region is also proposed. The algorithm proposed here is useful in analyzing and designing the robust PI^λD^μ controller for interval plant. An example is given to show how the presented algorithm can be used to compute all the parameters of a PI^λD^μ controller which stabilize a interval plant family.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

N-soliton solution of a lattice equation related to the discrete MKdV equation

We present an N-soliton solution of a lattice equation related to the discrete MKdV equation under an arbitrary boundary value at infinity.