- 8.2.1: Consider an M/M/1 queue with an arrival rate and the service rate ....
- 8.2.2: A group of telephone subscribers is observed continuously during a ...
- 8.2.3: Consider an M/M/1 queuing system in which the total number of jobs ...
- 8.2.4: The arrival of large jobs at a server forms a Poisson process with ...
- 8.2.5: Let the random variable Tk denote the holding time in state k of th...
- 8.2.6: Derive an expression for the frequency of entering state 0 (server ...
- 8.2.7: Define the perceived mean queue length [GEIS 1983] N = E[N2]/E[N] a...
Solutions for Chapter 8.2: The Birth-Death Process
Full solutions for Probability and Statistics with Reliability, Queuing, and Computer Science Applications | 2nd Edition
In hypothesis testing, a region in the sample space of the test statistic such that if the test statistic falls within it, the null hypothesis cannot be rejected. This terminology is used because rejection of H0 is always a strong conclusion and acceptance of H0 is generally a weak conclusion
Analysis of variance (ANOVA)
A method of decomposing the total variability in a set of observations, as measured by the sum of the squares of these observations from their average, into component sums of squares that are associated with speciic deined sources of variation
Binomial random variable
A discrete random variable that equals the number of successes in a ixed number of Bernoulli trials.
Bivariate normal distribution
The joint distribution of two normal random variables
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.
Central limit theorem
The simplest form of the central limit theorem states that the sum of n independently distributed random variables will tend to be normally distributed as n becomes large. It is a necessary and suficient condition that none of the variances of the individual random variables are large in comparison to their sum. There are more general forms of the central theorem that allow ininite variances and correlated random variables, and there is a multivariate version of the theorem.
Chi-square (or chi-squared) random variable
A continuous random variable that results from the sum of squares of independent standard normal random variables. It is a special case of a gamma random variable.
Any test of signiicance based on the chi-square distribution. The most common chi-square tests are (1) testing hypotheses about the variance or standard deviation of a normal distribution and (2) testing goodness of it of a theoretical distribution to sample data
Conditional probability mass function
The probability mass function of the conditional probability distribution of a discrete random variable.
An estimator that converges in probability to the true value of the estimated parameter as the sample size increases.
Continuous uniform random variable
A continuous random variable with range of a inite interval and a constant probability density function.
In the most general usage, a measure of the interdependence among data. The concept may include more than two variables. The term is most commonly used in a narrow sense to express the relationship between quantitative variables or ranks.
Discrete uniform random variable
A discrete random variable with a inite range and constant probability mass function.
Error sum of squares
In analysis of variance, this is the portion of total variability that is due to the random component in the data. It is usually based on replication of observations at certain treatment combinations in the experiment. It is sometimes called the residual sum of squares, although this is really a better term to use only when the sum of squares is based on the remnants of a model-itting process and not on replication.
The variance of an error term or component in a model.
Estimator (or point estimator)
A procedure for producing an estimate of a parameter of interest. An estimator is usually a function of only sample data values, and when these data values are available, it results in an estimate of the parameter of interest.
A subset of a sample space.
Exponential random variable
A series of tests in which changes are made to the system under study
Gamma random variable
A random variable that generalizes an Erlang random variable to noninteger values of the parameter r
A function that is used to determine properties of the probability distribution of a random variable. See Moment-generating function