Ping Cao

Mathematical modeling of software reliability testing with imperfect debugging

Software reliability testing is concerned with the quantitative relationship between software testing and software reliability. Our previous work develops a mathematically rigorous modeling framework for software reliability testing. However the modeling framework is confined to the case of perfect debugging, where detected defects are removed without introducing new defects. In this paper the modeling framework is extended to the case of imperfect debugging and two models are proposed. In the first model it is assumed that debugging is imperfect and may make the number of remaining defects reduce by one, remain intact, or increase by one. In the second model it is assumed that when the number of remaining defects reaches the upper bound, the probability that the number of remaining defects is increased by one by debugging is zero. The expected behaviors of the cumulative number of observed failures and the number of remaining defects in the first model show that the software testing process may induce a linear or nonlinear dynamic system, depending on the relationship between the probability of debugging introducing a new defect and that of debugging removing a detected defect. The second-order behaviors of the first model also show that in the case of imperfect debugging, although there may be unbiased estimator for the initial number of defects remaining in the software under test, the cumulative number of observed failures and the current number of remaining defects are not sufficient for precisely estimating the initial number of remaining defects. This is because the variance of the unbiased estimator approaches a non-zero constant as the software testing process proceeds. This may be treated as an intrinsic principle of uncertainty for software testing. The expected behaviors of the cumulative number of observed failures and the number of remaining defects in the second model show that the software testing process may induce a nonlinear dynamic system. However theoretical analysis and simulation results show that, if defects are more often removed from than introduced into the software under test, the expected behaviors of the two models tend to coincide with each other as the upper bound of the number of remaining defects approaches infinity.

Optimal dynamic pricing problem considering patient and impatient customers’ purchasing behaviour

We divide customers into two types according to their purchasing behaviour: one is impatient and the other is patient. An impatient customer will be immediately lost if his reservation price of the product is below the current price, while a patient customer will wait until the price drops below his reservation price and buy one item. The seller’s objective is to maximise his expected discounted overall revenue by dynamically pricing. We prove that the optimal pricing policy has a control limit structure, examine the impact of the proportion parameter on the seller’s expected discounted overall revenue and show that the limiting behaviour of the optimal pricing policy when there are little patient customers is the same as the optimal static pricing policy. We also show that the seller will make more profit if he partially knows the number of waiting customers by providing subscription service to customers. A set of numerical results illustrates the value of dynamic pricing and information with respect to problem parameters such as the arriving probability and the subscription probability.

Quantitative effects of software testing on reliability improvement in the presence of imperfect debugging

Software testing is essential for software reliability improvement and assurance. However, software testing is subject to imperfect debugging in the sense that new defects may be introduced into the software under test while detected defects are removed. The quantitative effects of software testing on software reliability improvement are obscure. In this paper we propose a Markov usage model to explore the quantitative relationships between software testing and software reliability in the presence of imperfect debugging. Several interesting quantities for software reliability assessment are derived and the corresponding upper and lower bounds are obtained.

Determining the conditions for reverse triage in emergency medical services using queuing theory

In emergency health care, there are situations where the less wounded are treated in preference to the more severely wounded, so called reverse triage. This may arise in situations such as war where soldiers are required to return to the battlefield as quickly as possible, or disaster situations where medical resources are limited in order to conserve resources for those likely to survive without requiring advanced medical care. This article is to study the reverse triage in time-critical systems, where conditions of patients may deteriorate while waiting for treatment. A queueing model with transfers and abandonments is developed to study the health care system. Using smoothed rate truncation method, sufficient conditions of reverse triage are derived, under which non-critical patients are treated with priority. For other conditions, the optimal policy is very complicated. We suggest a dynamic control policy according to the system state rather than any static priority policies.

Optimal Drift Rate Control and Impulse Control for a Stochastic Inventory/Production System

In this paper, we consider joint drift rate control and impulse control for a stochastic inventory system under long-run average cost criterion. Assuming the inventory level must be nonnegative, we prove that a

Optimal Control of an Inventory System With Joint Production and Pricing Decisions

\{(0,q^{\star},Q^{\star},S^{\star}),\{\mu^{\star}(x): x\in[0, S^{\star}]\}\}

Impacts of random capacity and fluctuating environment on inventory systems

policy is an optimal joint control policy, where the impulse control follows the control band policy

An optimal release policy for software testing process

(0,q^{\star},Q^{\star},S^{\star})

Robust Dynamic Selection of Tested Modules in Software Testing for Maximizing Delivered Reliability.

, that brings the inventory level up to

The optimal production and sales policy for a new product with negative word-of-mouth

q^{\star}