THE RANDOM WALK [IN CONTINUOUS TIME] AND ITS APPLICATION TO THE THEORY OF QUEUES
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连续时间随机游走及其在排队理论中的应用
- 研究问题
- 如何统一处理连续时间随机游走的不同边界条件，并将所得解用于有限容量单服务台、多服务台排队及扩散过程研究？
- 方法与思路
- 作者建立覆盖不同边界条件的统一理论，考察反射边界与容量极限、扩散过渡，并研究转移强度随位置分段变化的过程。
- 材料陈述的贡献
- 摘要称该理论包含既有结果并可处理其他相容边界条件，同时给出多服务台排队概率生成函数的拉普拉斯变换。
- 阅读关注点
- 应核对边界条件的相容性、随机游走与排队状态的对应关系、无限容量及扩散极限步骤，以及转移强度的具体设定。
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原文摘要
The random walk in continuous time has been considered by a number of authors for particular types of boundary conditions, e. g. the unrestricted random walk by Irwin (1937); the random walk with one absorbing or one reflecting barrier by Ledermann & Reuter (1954), and subsequently by Bailey (1954). The present paper gives a unified theory of these processes which includes previous work on the subject as special cases and readily yields the solution for any other consistent form of boundary conditions. Several of these are of interest in the theory of queues with exponential service and arrival times. The random walk with reflecting barriers at Oand N (§ 4) is the single server queue problem where the number of possible customers is limited to N (Morse, 1958). By letting N → ∞ we obtain the familiar single server queue problem solved previously in the papers by Ledermann & Reuter and by Bailey referred to above and also by Clarke (1956), Champernowne (1956) and Conolly (1958). The passage to a ‘diffusion’ process is also considered. The method given provides an alternative to the well-known method of images (Chandrasekhar, 1943; Bartlett, 1956). In § 5 we consider a more general process where the transition intensities λ and μ depend linearly on the particle‘s position in a bounded interval, and are constant outside this interval. This enables us to give the Laplace transform of the probability generating function for the N server queue, a problem recently considered by Karlin & McGregor (1958).