By J.A. White (Auth.)
Publication via White, John A., and so forth
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Extra resources for Analysis of Queueing Systems
10 Find the mean and variance of the Poisson random variable. The probability mass function of the Poisson random variable is p(x) = ^ e - * T , x = 0,1,2,... 44 2. 69) *=o x! χ=ο x = ! (χ - 1)! Let y = χ — 1. e-Ar 2= σ + 2 χ( Α Ώ ! 1. 11 A company places orders for raw materials t weeks prior to scheduled production. The time X taken to receive an order after it has been placed is exponentially distributed with parameter A. If the order arrives prior to production, χ < r, an inventory carrying cost of 1500(r — x) is Probability Theory 45 incurred.
However, their applications to queueing theory are limited. Even so, they are of use in analyzing queueing systems through simulation and in data analysis prior to the development of queueing models. These applications will be demonstrated when we discuss data collection and analysis and in the chapter on the simulation of queueing systems. Example 25 An income tax consultant works by himself. He spends an average of about 30 minutes with each client. However, the time spent with each client is exponentially distributed.
74) holds no matter how small or large Τ is, provided Τ > 0. For Τ = 0, p(x) = 09 χ =1,2,... 77) and p(x) = e-W) =1, and the fifth assumption is verified. The third assumption states that if X has a Poisson distribution, then no two events may occur at the same point in time. Suppose we consider the occurrence of χ events in a time interval of length Δί. Then P( ) = ~T ~ > x e x = XM 0,1,2,... Now _ £ (178) * ) - ^ Σ ^ - Σ ( - ι ^ <™> Therefore, Χ! =ί 0 II = 0i Xlll Now let us define a number 0(Δί) > 0 such that if | k | < 0(Δί), k may be considered zero for all practical purposes.
Analysis of Queueing Systems by J.A. White (Auth.)