Linear Cost Structure Research Articles

Batch arrival queue with N-policy and single vacation

We consider an M x / G/1 queueing system with N-policy and single vacation. As soon as the system becomes empty, the server leaves the system for a vacation of random length V. When he returns from the vacation, if the system size is greater than or equal to predetermined value N(threshold), he beings to serve the customers. If not, the server waits in the system until the system size reaches or exceeds N. We derive the system size distribution and show that the system size distribution decomposes into two random variables one of which is the system size of ordinary M x / G/1 queue. The interpretation of the other random variable will also be provided. We also derive the queue waiting time distribution of an arbitrary customer. Finally we develop a procedure to find the optimal stationary operating policy under a linear cost structure.

THE OPTIMAL (r,S) POLICY IN A SINGLE ITEM PRODUCTION/INVENTORY SYSTEM WITH SETUP TIMES

In this paper, the (r,S) policy is considered for production/inventory systems where a single type items are produced on an item by item basis by a single production facility. The (r,S) policy considered in this paper is a pull type threshold policy which is very useful in situations where the time and cost required to setup the production facility are relatively high. The demand for the item is assumed to arrive according to a Poisson process. The processing time required to produce an item is assumed to follow an arbitrary distribution. When an item is demanded, one of the items in the inventory is delivered to the customer and the kanban on the item is removed. The removed kanban is immediately transmitted to the production facility. At the instant r kanbans are accumulated at the production facility, the operator turns the production facility on, which takes a random amount of time. In the production period, items are produced one by one and whenever each item is produced, a kanban is attached to it. When there are no kanbans at the production facility, the machine is shut off and a non-production period begins, which lasts until the number of kanbans accumulated at the production facility is raised back to r. In this paper, assuming a linear cost structure, an efficient search procedure is developed to find the optimal threshold value r as well as the optimal number of kanbans S, which minimizes the expected cost incurred per unit time.

Analysis of the Mx/G/1 queue by N-policy and multiple vacations

We consider an Mx/G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary Mx/G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure.

Analysis of the M x/G/1 queue by N-policy and multiple vacations

We consider an Mx /G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary Mx /G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure.

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

We consider aMX/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 theMX/G/1 queueing system withoutN-policy and the other one has the probability generating function ∑ j=0 N=1 π j zj/∑ j=0 N=1 π j , in which πj 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.

Optimal dispatching of an infinite capacity shuttle with compound poisson arrivals: Control at a single terminal

We consider the optimal control of an infinite capacity shuttle which transports passengers between two terminals. Passengers arrive at each of two terminals according to independent compound Poisson processes. The travel time from one terminal to the other one is a random variable following an arbitrary distribution. The dispatcher can hold the shuttle at only one of the terminals. Thus, when the shuttle is dispatched at terminal 1, it transports all the passengers waiting at terminal 1 to terminal 2 and instantaneously picks up the passengers at terminal 2 and returns to terminal 1. Assuming that the dispatcher has complete information about the number of passengers waiting at each terminal, we consider the following control limit policy: dispatch the shuttle if, and only if, the total number of passengers waiting at both terminals is at least as large as some prespecified control limit, m. In this paper, under a linear cost structure, we provide an explicit expression for the expected cost per unit time as a function of the control value, m, and present a simple and efficient method for computing the optimal control value.

Stochastic analysis of an Erlangian arrival with linear cost structure

In this paper a control policy for an Erlangian arrival has been studied. Here, the server goes on vacation when the system becomes empty, returning to an operative state after a random idle period lapses, or the queue length becomes N, whichever is earlier. The expressions for various operational characteristics have been obtained. Numerical results have been computed and some of these are graphed. Graphical interpretation of linear cost structure has been discussed.

Joint product signals of quality

The relationship between product quality, signals, and the firm's optimal pricing policy has been given much attention in economics. The literature is extended in this paper by considering the signaling problem of a firm that jointly produces two commodities—one of known quality to consumers (a search good) and one of unknown quality (an experience good). The model presented employs a stylized reputation function, a linear cost structure, and linear demand schedules to produce two interesting insights. First, the search good's price can potentially be used as a signal of the quality of the experience good. Second, the price of a search good will depend upon whether it is jointly produced with another search good or an experience good or whether it is produced in isolation by a single product firm. Furthermore, evidence from a paper on gasoline pricing seems to support this contention.

The shuttle dispatch problem with compound Poisson arrivals: Controls at two terminals

We consider the control of an infinite capacity shuttle which transports passengers between two terminals. The passengers arrive at each terminal according to a compound Poisson process and the travel time from one terminal to the other is a random variable following an arbitrary distribution. The following control limit policy is considered: dispatch the shuttle at terminali, at the instant that the total number of passengers waiting at terminali reaches or exceeds a predetermined control limitm i . The objective of this paper is to obtain the mean waiting time of an arbitrary passenger at each terminal for given control valuesm 1 andm 2. We also discuss a search procedure to obtain the optimal control values which minimize the total expected cost per unit time under a linear cost structure.

A short note on the poisson input queuing system with start-up time and under control-operating policy

The control policy for the M/G/1 queuing system in which the server is started-up when the accumulated number of customers reaches a predetermined value m and serves the system until it is empty. The time required to start up the server follows a general distribution. The mean waiting time of an arbitrary customer for a given value of m is derived and the optimal control policy is presented under a linear cost structure.

State reduction in a dependent demand inventory model given by a time series

The dependent demand inventory model given by the time series with a stochastic trend and seasonality was considered. A two-state variable dynamic programming problem was reduced to the one-state variable problem under linear cost structure. The demand distribution was shown to be normal with the nonstationary mean and the constant variance. The optimal ordering levels are characterized by single critical numbers.

Control Policies for the MX/G/1 Queueing System

The MX/G/1 queueing system is studied under the following two situations: (1) At the end of a busy period, the server is turned off and inspects the length of the queue every time an arrival occurs. When the queue length reaches, or exceeds, a pre-specified value m for the first time, the server is turned on and serves the system until it is empty. (2) At the end of a busy period, the server takes a sequence of vacations, each for a random amount of time. At the end of each vacation, he inspects the length of the queue. If the queue length is greater than, or equal to, a pre-specified value m at this time, he begins to serve the system until it is empty. For both cases, the mean waiting time of an arbitrary customer for a given value of m is derived, and the procedure to find the stationary optimal policy under a linear cost structure is presented.

A stochastic optimal control approach to a class of production and inventory problems

A comprehensible and unified system control approach is presented to solve a class of production/inventory smoothing problems. A nonstationary, non-Gaussian, finite-time linear optimal solution with an attractive computation scheme is obtained for a general quadratic and linear cost structure. A complete solution to a classical production/inventory control problem is given as an example. A general solution to the discrete-time optimal regulator with arbitrary but known disturbance is provided and discussed in detail. A computationally attractive closed-loop suboptimal scheme is presented for problems with constraints or nonquadratic costs. Implementation and interpretation of the results are discussed.

Performance of an approximation to continuous review (s, S) policies under compound renewal demand†

SUMMARY Performance of an approximation to continuous review (s,S) inventory policies under constant lead time, full backlogging, compound renewal demand, and a linear cost structure is examined. The approximations are based on a lower bound for the optimal order-up-to level and the corresponding best reorder point. It is shown that the approximate policy is unique and that it performs better under relatively higher shortage costs, longer lead times, and lower set-up costs. Both exact and approximate policies are computed and the corresponding objective function values are compared under a large number of different parameter settings. The average relative approximation error is found to be 2·23% with a 27-fold average reduction in computing times.

Optimal Entering Rules for a Customer with Wait Option at an M/G/1 Queue

A “smart” customer arrives at an M/G/1 queue. While every other arrival joins the system unconditionally, our customer is allowed to choose among three alternatives: (i) he may Enter the queue and stay there until his service is completed, (ii) he may Leave the system right away, or (iii) he may Wait outside the queue. If he Enters or Leaves the system, his decision is final and no further actions are taken. If he chooses to Wait, he makes a new decision at the next service completion where he may, again, select one of the three options: Enter, Leave, or Wait. For a linear cost structure we show that for any n-period horizon (0 < n < ∞) the individual customer's optimal strategy is a 3-region (possibly degenerate) policy by which he Enters a small queue, Leaves a large one and Waits when the queue is of an intermediate size. We also give a necessary and sufficient condition for the Wait option to exist. The special M/M/1 queue is further analyzed by allowing the Waiting customer to make decisions at instants of customers' arrivals as well as at instants of service completion. In this case, too, the optimality of the 3-region policy is derived and the “smart” customer dichotomy is recognized as the Gambler's Ruin problem. Computational procedures are developed and numerical results are presented for commonly used service time distributions.

Optimal Operating Policy for a Random Additional Service Facility

This paper studies the steady state behaviour of a Markovian queue wherein there is a regular service facility serving the units one by one. A search for an additional service facility for the service of a group of units is started when the queue length increases to K (0 < K < L), where L is the maximum waiting space. The search is dropped when the queue length reduces to some tolerable fixed size L - N. The availability time of an additional service facility is a random variable. The model is directed towards finding the optimal operating policy (N,K) for a queueing system with a linear cost structure.

Analysis and Design of Loop Service Systems via a Diffusion Approximation

In a loop service system the server cyclically serves a fixed number of peripheral stations. In this paper we consider an operating rule where the server is allowed to service a particular station for a fixed length of time each cycle. Using a diffusion approximation, we obtain expressions for the distribution of the steady state load as well as the expected waiting time at a particular peripheral station. These results are then used to determine the optimal design for the system under a linear cost structure.

Optimal production growth for the machine loading problem

AbstractThis paper investigates a production growth logistics system for the machine loading problem (generalized transportation model), with a linear cost structure and minimum levels on total machine hours (resources) and product types (demands). An algorithm is provided for tracing the production growth path of this system, viz. in determining the optimal machine loading schedule of machines for product types, when the volumes of (i) total machine hours, and (ii) the total amount of product types are increased either individually for each total or simultaneously for both. Extensions of this methodology, when (i) the costs of production are convex and piecewise linear, and (ii) when the costs are nonconvex due to quantity discounts, and (iii) when there are upper bounds for productions are also discussed. Finally, a “goal‐programming” production growth model where the specified demands are treated as just goals and not as absolute quantities to be satisfied is also considered.

Dynamic Scheduling of a Multiclass Queue: Discount Optimality

We consider a single-server queuing system with several classes of customers who arrive according to independent Poisson processes. The service time distributions are arbitrary, and we assume a linear cost structure. The problem is to decide, at the completion of each service and given the state of the system, which class (if any) to admit next into service. The objective is to maximize the expected net present value of service rewards received minus holding costs incurred over an infinite planning horizon, the interest rate being positive. One very special type of scheduling rule, called a modified static policy, simply enforces a (nonpreemptive) priority ranking except that certain classes are never served. It is shown that there is a modified static policy that is optimal, and a simple algorithm for its computation is presented.

Determining optimal growth paths in logistics operations

AbstractThis paper considers a logistics system modelled as a transportation problem with a linear cost structure and lower bounds on supply from each origin and to each destination. We provide an algorithm for obtaining the growth path of such a system, i. e., determining the optimum shipment patterns and supply levels from origins and to destinations, when the total volume handled in the system is increased. Extensions of the procedure for the case when the costs of supplying are convex and piecewise linear and for solving transportation problems that are not in “standard form” are discussed. A procedure is provided for determining optimal plant capacities when the market requirements have prespecified growth rates. A goal programming growth model where the minimum requirements are treated as goals rather than as absolute requirements is also formulated.