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This paper treats an M/M/1 retrial queue with constant retrial times. If the server is busy at the arrival epoch, the arriving customer decides to join the retrial orbit with probability q or balk with probability 1-q. Only the customer at the head of the orbit is permitted to access the server. Upon retrial, the customer immediately receives his service if the server is idle; otherwise, he may enter the orbit again or leave the system because of impatience. First, we give the performance analysis for this retrial queue and give some important performance indices. Second, based on a natural reward-cost structure, we analyze the Nash equilibrium customers’ joining strategies and give some numerical examples.

To address the issue of whether there exists determinism in a two-phase flow system, we first conduct a gas-liquid two-phase flow experiment to collect the flow pattern fluctuation signals. Then we investigate the determinism in the dynamics of different gas-liquid flow patterns by calculating the number of missing ordinal patterns associated with the partitioning of the phase space. In addition, we use the recently proposed stretched exponential model to reveal the flow pattern transition behavior. With the joint distribution of two fitted parameters, which are the decay rate of the missing ordinal patterns and the stretching exponent, we systematically analyze the flow pattern evolutional dynamics associated with the flow deterministic characteristics. This research provides a new understanding of the two-phase flow pattern evolutional dynamics and broader applications in more complex fluid system is suggested. The two-phase flow, such gas and liquid, is a frequently encountered dynamical system in many fields, such as nuclear, petrochemical, biochemical, hydraulic, and others. A fundamental problem of the two-phase flow system is the presence of both determinism and stochasticity in its dynamics. Understanding the two-phase flow characteristics and its deterministic evolutionary dynamics is of importance and of continuing interest for the flow control system optimization. For that, we need to characterize the system determinism, what we accomplish by partitioning the phase space, assigning symbols to each partition, and then counting the number of missing ordinal patterns (NMP) during the system's evolution. This is a sensitive and a reliable index to characterize the system's determinism. We design an experimental gas-flow facility and carry out the experiments by exploring parameter space of possible flow conditions. The collected fluctuation data are then employed to assess determinism using our NMP method. We further investigate the NMP scaling behavior of the flow fluctuations to demonstrate the diverse gasliquid two-phase flow evolutionary dynamics. The results of this work show that the gas-liquid two-phase flow deterministic dynamics is associated with a phase morphology distribution, called the flow patterns. The methods used in this work are expected to apply in more complex fluid systems such as multi-phase flow and microfluidics. I. INTRODUCTION Gas-liquid two-phase flow widely exists in industrial processes such as crude oil production, nuclear-power generation, fluidization, and so on. How to clearly characterize the coma) Electronic mail:

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