1. 一计算机系统有一个操作单元和一个备用单元。操作单元损坏的时间服从均值为1000的指数分布。如果操作单元坏了，立即启用备用单元。备用单元在等待状态时损坏的时间服从均值为2000的指数分布，在工作状态时损坏的时间服从均值为500的指数分布。每个损坏的单元都需要修理，并已知只有一个修理员，操作单元的修理时间服从均值为50的指数分布，备用单元的修理时间服从均值为30的指数分布，并且操作单元优先于备用单元修理。求：

（1） 状态转移矩阵；

（2） 长时间后，系统处于工作状态的比例；

（3） 系统连续工作2000单位时间而不发生损坏的概率。

2. 在一个存贮系统中，产品的需求为参数为λ的泊松过程 。得不到直接满足的需求将会失去。库存获得补充的机会是参数为μ的泊松过程，并且只有在库存为零时才去订货。每个订单的产品数量是Q，前置期为零（即货物可以立即到达）。每个订单的成本为K，单位时间里单位产品的存贮费是h，每失去一单位产品销售的成本为π。求长时间后，系统在单位时间里的成本。

问题原文：

1. A Computer system has one operating unit and one standby unit. The operating unit has an exponential time with mean 1000 to failure. If the operating unit fails, it is replaced immediately by the standby unit. The standby unit has an exponential time with mean 2000 to failure at the standby state but has an exponential time with mean 500 to failure at the operating state. Each failed unit is repaired and there is one repairman. The repair times are distributed exponentially and the mean times are 50 for the operating unit and 30 for the standby unit. The operating unit has priority in repair over the standby unit. Obtain:

(1) The infinitesimal generator.

(2) The long run average portion that the computer system works.

(3) The probability that the computer system works to 2000 without system failure.

2. In an inventory system for a single product the demand process is a Poisson process with rate λ. Each demand which is not satisfied directly is lost. Opportunities to replenish the inventory occur according to a Poisson process with rate μ. A replenishment order can only be made when the inventory equals zero. Each replenishment order consists of Q. The lead time is zero. There is a fixed cost of K per replenishment order, a holding cost of h per unit inventory per unit time and a lost sales cost of π per unit demand lost. Determine the long-run average cost per unit time.