Steady-state Probabilities Of System Research Articles

Inventory rationing on a one-for-one inventory model for two priority customer classes with backorders and lost sales

In this study, we are primarily motivated by the research problem of recognizing heterogeneous customer behavior towards waiting for order fulfillment under the threshold rationing policy (also known as the critical level policy), and aim to find its effect on system stock levels and performance measures. We assume a continuous review one-for-one ordering policy with generally distributed lead times. In the first model, we consider the case in which the low-priority customer class exhibits zero patience for waiting if the demand is not satisfied immediately (a lost sale), whereas the demand of the high-priority customer class can be backordered. This is the first study in the literature to consider this model. We provide an exact analysis for the derivation of the steady-state probability distribution and the average infinite horizon cost per unit time. We then develop an efficient optimization procedure to minimize the average expected cost rate. We also determine the forms of the optimal solutions for the two service level optimization models that are common in practice. In the second model, we study the opposite case in which the high-priority customer class exhibits zero patience for waiting. We establish a theoretical basis for the rationale of using the Continuous-Time Markov Chain (CTMC) approach as an approximation. We show that under certain assumptions, the steady-state probabilities of the system with generally distributed lead times are identical to the steady-state probabilities of the CTMC system with the same mean. This result enables us to link the dynamics of the studied model to the CTMC model, which may open new doors for future research.

Inventory Rationing on a One-for-One Inventory Model with Backorders and Lost Sales

In this study, we are primarily motivated by the research problem of recognizing heterogeneous customer behavior towards waiting for order fulfillment under the 'threshold rationing policy' (also known as the 'critical level policy'), and aim to find its effect on system stock levels and performance measures. We assume a continuous review one-for-one ordering policy with generally distributed lead times. In the first model, we consider the case in which low-priority customer class exhibits zero patience for waiting if the demand is not satisfied immediately (a lost sale), while the demand of high-priority customer class can be backordered. This is the first study in the literature to consider this model. We provide an exact analysis for the derivation of the steady-state probability distribution and the average infinite horizon cost per unit time. We then develop an efficient procedure to minimize the average expected cost rate by deriving bounds on the optimal cost, which reduces the enumeration space considerably. In the numerical study, we show the relative savings achieved by implementing the proposed threshold rationing policy over some of the traditional industry practices. In the second model, we study the opposite case in which the high-priority customer class exhibits zero patience for waiting. The existing study in the literature uses simulation to demonstrate that the performance of the critical level policy is robust with respect to the form and variability of the lead time distribution provided the mean lead times are identical. Then it proposes using the Continuous-Time Markov Chain (CTMC) equivalent of the model as an approximation. In this study, we establish a theoretical basis for the rationale of using the CTMC approach as an approximation. We show that under certain assumptions the steady-state probabilities of the system with generally distributed lead times are identical to the steady-state probabilities of the CTMC system with the same mean. This result enables us to link the dynamics of the studied model to the CTMC model, which then allows us to provide a theoretical explanation to the empirically observed phenomenon that why the steady-state probabilities and performance measures are near-insensitive to the form and variability of the lead time distribution as long as mean lead times are identical.

Estimating the deviation of steady-state probabilities of system GI / G / m / r from state probabilities of system M / M / m / r by analytical-statistical method

An algorithm is proposed to model "corrections" to characteristics of a general model where exponential distributions are substituted with nonexponential ones. This algorithm is used to evaluate the deviation of steady-state probabilities of system GI/G/m/r from steady-state probabilities of system M/M/m/r. A numerical example is given.