On the discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper examines a discrete-time Geo/G/1 queue, where the server may take at most J − 1 vacations after the essential vacation. In this system, messages arrive according to Bernoulli process and receive corresponding service immediately if the server is available upon arrival. When the server is busy or on vacation, arriving messages have to wait in the queue. After the messages in the queue are served exhaustively, the server leaves for the essential vacation. At the end of essential vacation, the server activates immediately to serve if there are messages waiting in the queue. Alternatively, the server may take another vacation with probability p or go into idle state with probability (1 − p) until the next message arrives. Such pattern continues until the number of vacations taken reaches J. This queueing system has potential applications in the packet-switched networks. By applying the generating function technique, some important performance measures are derived, which may be useful for network and software system engineers. A cost model, developed to determine the optimum values of p and J at a minimum cost, is also studied.     

The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption

Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0 ? p ? 1) or continue the vacation with probability 1 − p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time.     

The randomized threshold for the discrete-time Geo/G/1 queue

This paper discusses a discrete-time Geo/G/1 queue, in which the server operates a random threshold policy, namely 〈p, N〉 policy, at the end of each service period. After all the messages are served in the queue exhaustively, the server is immediately deactivated until N messages are accumulated in the queue. If the number of messages in the queue is accumulated to N, the server is activated for services with probability p and deactivated with probability (1 − p). Using the generating functions technique, the system state evolution is analyzed. The generating functions of the system size distributions in various states are obtained. Some system characteristics of interest are derived. The long-run average cost function per unit time is analytically developed to determine the joint optimal values of p and N at a minimum cost.     

Symmetric duality with (p, r) − ρ − (η, θ)-invexity

In this paper we introduce a new type of generalized invex function, called (p, r) − ρ − (η, θ)-invex function and study symmetric duality results under these assumptions. In our study the nonnegative orthants for the constraints are replaced by closed convex cones and their polars. We establish weak and strong duality theorems under (p, r) − ρ − (η, θ)-invexity assumptions for the symmetric dual problems. We also give many examples to justify our results.     

Fractal systems of central places based on intermittency of space-filling

The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d = 2 indicating no intermittency in the spatial distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D < d = 2 indicative of spatial intermittency. Thus the conventional central place models are converted into fractal central place models. If we further integrate the chance factors into the improved central place fractals, the theory will be able to explain the real patterns of urban places very well. As empirical analyses, the US cities and towns are employed to verify the fractal-based models of central places.     

Optimal (d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs

This paper considers a finite buffer M/M/c queueing system in which servers are unreliable and follow a (d, c) vacation policy. With such a policy, at a service completion instant, if the number of customers is reduced to c − d (c > d), the d idle servers together take a vacation (or leave for a random amount of time doing other secondary job). When these d servers return from a vacation and if still no more than c − d customers are in the system, they will leave for another vacation and so on, until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. This study is motivated by the fact that some practical production and inventory systems or call centers can be modeled as this finite-buffer Markovian queue with unreliable servers and (d, c) vacation policy. Using the Markovian process model, we obtain the stationary distribution of the number of customers in the system numerically. Some cost relationships among several related systems are used to develop a finite search algorithm for the optimal policy (d, c) which maximizes the long-term average profit. Numerical results are presented to illustrate the usefulness of such a algorithm for examining the effects of system parameters on the optimal policy and its associated average profit.     

Synchronous working vacation policy for finite-buffer multiserver queueing system

This paper presents modeling and analysis of unreliable Markovian multiserver finite-buffer queue with discouragement and synchronous working vacation policy. According to this policy, c servers keep serving the customers until the number of idle servers reaches the threshold level d; then d idle servers take vacation altogether. Out of these d vacationing servers, d_W servers may opt for working vacation i.e. they serve the secondary customers with different rates during the vacation period. On the other hand, the remaining d − d_W = d_V servers continue to be on vacation. During the vacation of d servers, the other e = c − d servers must be present in the system even if they are idle. On returning from vacation, if the queue size does not exceed e, then these d servers take another vacation together; otherwise start serving the customers. The servers may undergo breakdown simultaneously both in regular busy period and working vacation period due to the failure of a main control unit. This main unit is then repaired by the repairman in at most two phases. We obtain the stationary performance measures such as expected queue length, average balking and reneging rate, throughput, etc. The steady state and transient behaviours of the arriving customers and the servers are examined by using matrix analytical method and numerical approach based on Runge-Kutta method of fourth order, respectively. The sensitivity analysis is facilitated for the transient model to demonstrate the validity of the analytical results and to examine the effect of different parameters on various performance indices.     

Analysis of the queue-length distribution for the discrete-time batch-service Geo/G/1/K queue

In this paper, we consider a discrete-time finite-capacity queue with Bernoulli arrivals and batch services. In this queue, the single server has a variable service capacity and serves the customers only when the number of customers in system is at least a certain threshold value. For this queue, we first obtain the queue-length distribution just after a service completion, using the embedded Markov chain technique. Then we establish a relationship between the queue-length distribution just after a service completion and that at a random epoch, using elementary ‘rate-in = rate-out’ arguments. Based on this relationship, we obtain the queue-length distribution at a random (as well as at an arrival) epoch, from which important performance measures of practical interest, such as the mean queue length, the mean waiting time, and the loss probability, are also obtained. Sample numerical examples are presented at the end.     

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.     

Economic application in a finite capacity multi-channel queue with second optional channel

We consider a finite capacity M/M/R queue with second optional channel. The interarrival times of arriving customers follow an exponential distribution. The service times of the first essential channel and the second optional channel are assumed to follow an exponential distribution. As soon as the first essential service of a customer is completed, a customer may leave the system with probability (1 − θ) or may opt for the second optional service with probability θ (0 ? θ ? 1). Using the matrix-geometric method, we obtain the steady-state probability distributions and various system performance measures. A cost model is established to determine the optimal solutions at the minimum cost. Finally, numerical results are provided to illustrate how the direct search method and the tabu search can be applied to obtain the optimal solutions. Sensitivity analysis is also investigated.     

Triple Solutions for Second-Order Three-Point Boundary Value Problems

We establish the existence of at least three positive solutions to the second-order three-point boundary value problem, u″ + f(t, u) = 0, u(0) = 0, αu(η) = u(1), where η: 0 lt; η < 1, 0 < α < 1/η, and f: [0, 1] × [0, ∞) → [0, ∞) is continuous. We accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works.     

A preemptive discrete-time priority buffer system with partial buffer sharing

This paper considers a Geo/Geo/1 discrete-time queue with preemptive priority. Both the arrival and service processes are Bernoulli processes. There are two kinds of customers: low-priority and high-priority customers. The high-priority customers have a preemptive priority over low-priority customers. If the total number of customers is equal or more than the threshold (k), the arrival of low-priority customers will be ignored. Hence the system buffer size is finite only for the low-priority customers. A recursive numerical procedure is developed to find the steady-state probabilities. With the aid of recursive equations, we transform the infinite steady-state departure-epoch equations set to a set of (k + 1) × (k + 2)/2 linear equations set based on the embedded Markov Chain technique. Then, this reduced linear equations set is used to compute the steady-state departure-epoch probabilities. The important performance measures of the system are calculated. Finally, the applicability of the solution procedure is shown by a numerical example and the sensitivity of the performance measures to the changes in system parameters is analyzed.     

Analysis of the adaptive MMAP[K]/PH[K]/1 queue: A multi-type queue with adaptive arrivals and general impatience

In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability a_i,k if the current workload is between threshold i − 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected.     

Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X(t) are obtained when the random variable ζ₁, which is describing a discrete interference of chance, has a triangular distribution in the interval [s, S] with center (S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a ≡ (S − s)/2 → ∞. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a.     

Inventory models for perishable items with inventory level dependent demand rate

This paper presents inventory models for perishable items with inventory level dependent demand rate. The models with and without backlogging are studied. In the backlogging model, it is assumed that the backlogging rate is dependent on the waiting time and the amount of products already backlogged simultaneously. Two cases that holding inventory is profitable or not are studied, respectively. The smallest shelf space to ensure shortage not occur when holding inventory is not profitable is obtained. In the model without backlogging, it is assumed that the remaining stock at the end of the inventory cycle is disposed of with salvage value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solution of these models are investigated. At last, some numerical examples are presented to illustrate the effectiveness of the proposed model. The model in this paper is generalization of present ones. In particularly, the model is reduced to Padmanabhan and Vrat’s when δ₁ = 0, and Dye and Ouyang’s when δ₂ = 0. If S = s and δ₂ = 0, it is Chang, Goyal and Teng’s model.     

Vacation policies for machine repair problem with two type spares

This paper studies the machine repair problem consisting of M operating machines with two types of spare machines (S = S₁ + S₂), and R servers (repairmen) who leave for a vacation of random length when there are no failed machines queuing up for repair in the repair facility. At the end of the vacation the servers return and operate two vacation policies. First, the servers take vacations repeatedly until they find the repair facility has at least one waiting failed machine in the queue. Second, the servers do not take a vacation again and remain idle until the first arriving failed machine arrives, which starts a busy period in the repair facility. For both policies, the servers have two service rates for repair-slow and fast. The matrix geometric theory is used to find the steady-state probabilities of the number of failed machines in the system as well as the performance measures. Some special cases are given. A direct search algorithm is used to simultaneously determine the optimal values of the number of two types of spares and the number of servers while maintaining a minimum specified level of system availability.     

Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy

In 2007, Huang proposed the optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy, in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides its customer a permissible delay period N (with N < M). In this paper, we extend his EPQ model to complement the shortcoming of his model. In addition, we relax the dispensable assumptions of N < M and others. We then establish an appropriate EPQ model to the problem, and develop the proper theoretical results to obtain the optimal solution. Finally, a numerical example is used to illustrate the proposed model and its optimal solution.     

Distributive equations of implications based on nilpotent triangular norms

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T(y, z)) = T(I(x, y), (x, z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0, y), y ∈ [0, 1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, y) = I(N(y), N(x)).     

M/(G1, G2)/1 retrial queue under Bernoulli vacation schedules with general repeated attempts and starting failures

This paper investigates a batch arrival retrial queue with general retrial times, where the server is subject to starting failures and provides two phases of heterogeneous service to all customers under Bernoulli vacation schedules. Any arriving batch finding the server busy, breakdown or on vacation enters an orbit. Otherwise one customer from the arriving batch enters a service immediately while the rest join the orbit. After the completion of two phases of service, the server either goes for a vacation with probability p or may wait for serving the next customer with probability (1 − p). We construct the mathematical model and derive the steady-state distribution of the server state and the number of customers in the system/orbit. Such a model has potential application in transfer model of e-mail system.     

Optimal control of the N policy M/G/1 queueing system with server breakdowns and general startup times

This paper deals with an N policy M/G/1 queueing system with a single removable and unreliable server whose arrivals form a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. When the queue length reaches N(N ? 1), the server is immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing service until there are no customers in the system. We analyze various system performance measures and investigate some designated known expected cost function per unit time to determine the optimal threshold N at a minimum cost. Sensitivity analysis is also studied.