K0x0 and y0 k0yo

Equation 7.64 can then be rewritten as where N and D are the matrices N and D with their last columns deleted. Note that (7.66) yields numerical values for and In terms of yo, (7.63) becomes Since the individual probabilities must sum to unity, we then have o + yoQsjE r fjtf o + yoQs l- H Thus, we may determine k0 from (7.67). After finding k0, we may substitute its value into (7.65) to determine xo and yo, noting that xq contains the vectors We may then use yo in (7.63) to...

Rate Matrix Computation via Generalized State Space Methods

An interesting alternative for determining the rate matrix is the so-called generalized state-space approach discussed in Akar, Oguz, and Sohraby 1998 . For the case of a QBD process, we shall see that a natural state vector for the system of balance equations is found by concatenating two level probability vectors. Specifically, we will define in a natural way the state vector where and are two successive level probability vectors. Again, in a very straightforward manner we will see that...

Ml Po

Finally, upon dividing the numerator and denominator of the last equation by we obtain the same result as before for the probability generating function. That is, The remainder of the solution is as before. I Exercise 3.29 Use (3.66) to find E n andE n2 . 3-1 Messages arrive to a statistical multiplexing system according to a Poisson process having rate Message lengths, denoted by are specified in octets, groups of 8 bits, and are drawn from an exponential distribution having mean Messages are...

TTdj lim Phdk j

Then, the random process is a discrete parameter Markov chain defined on the nonnegative integers and is called an embedded Markov chain. In particular, the process hd(k),k 1,2, is called the occupancy process embedded at points immediately following customer departure. For the stationary probability vector, exists and where Vd is the one-step transition probability matrix (Ross 1989 ) for the embedded Markov chain (fc), k 1,2, and e is the column vector in which each element is unity. For the...

CDMABased Cellular Data

High data rate transmission based on frame-oriented time division multiplexing has been proposed as a paradigm for forward-link transmission in CMDA-based cellular systems (Bender 2000 ). In such a system, the capac- Figure 1.7. Queue length survivor function for an 8-to-1 multiplexing system at a traffic intensity of 0.9 with average run length as a parameter. ity of a frame depends upon the signal plus noise to interference ratio (SINR) of the target mobile receiver, which is dependent upon...

Random Variables

In this section, we first provide a formal definition of the term random variable, and then we discuss distributions of random variables. Next, we discuss characterization of random variables according to the form of their distribu tions. We then discuss probability mass functions and probability density functions for random variables. Next we discuss computation of expectation of functions of random variables. Finally, we discuss computation of the distribution of sums of random variables....

N. Akar Sohraby 2012 Queue

Survivor functions for occupancy distributions for statistical multiplexing system with 0.5 to 1.0 speed conversion at p 0.9. For fixed and 1 - a , we can then solve for The average message length is eight packets, so 1 - a 0.125. This, then, completely specifies the parameters for the model. The recursion of 7.42 with C0 l as defined in 7.45 is then used to determine C l and its stationary probability vector, k, is determined by solving the system k0 kqICqo, oe 1. Then, I is...

Poisson Process

The characterization of arrival processes for many queueing systems as Poisson has a solid physical basis, as was first discovered by A. K. Erlang during the 1910's. The Poisson assumption can reduce the analytical complexity of a problem and lead to easily obtained and useful results, but the same assumption may also render the analysis useless. As seen in the examples presented in Chapter 1, while the Poisson characterization is often appropriate, there are many cases in which the Poisson...

Info

4We will use this definition for e in the remainder of the text. The proofs of these theorems are left as exercises. I Exercise 3.17 Prove Theorem 3.5. Exercise 3.19 Let K 1. Use Definition 2 of the Poisson process to write an equation of the form Show that the eigenvalues of the matrix Q are real and nonnegative. Solve the equation for Po t , P t and show that they converge to the solution given in Example 3.2 regardless of the values o 0 , Pi 0 . Hint First, do a similarity transformation on...

Exponential Distribution

Certain ideas and concepts from the theory of stochastic processes are basic in the study of elementary queueing systems. Perhaps the most important of these are the properties of the exponential distribution and the Poisson process. The purpose of this and the next section is to discuss these and related concepts. We begin with a definition of the memoryless property of a random variable and then relate this to the exponential distribution. Much of the literature and results in stochastic...

Ergodic Occupancy Distributions via Generalized State Space Approach

As discussed in Section 5.1, the process gn, n gt 1 , which denotes the number of customers left in the system by the nth departing customer, is a Markov chain embedded at points of customer departure. Our objective is to present a linear algebra-based approach for obtaining the distribution of the number of customers left in the system by a departing customer for a variation of the M G 1 system. We specifically consider the cases in which the probability generating function of the number of...

Cellular Telephony

In an analog cellular communication system, there are a total of 832 available frequencies, or channels. These are typically divided between two service vendors so that each vendor has 416 channels. Of these 416 channels, 21 are set aside for signalling. A cellular system is tessellated, meaning that the channels are shared among a number of cells, typically seven. Thus, each cell has about 56 channels. In order to get a feel for where cell cites should be placed, the vendor would like to...

Busy Period for the MG1 Queueing System

In this section, we will determine the Laplace-stieltjes transform for the distribution of the length of the busy period for the M G 1 queueing system. As before, we let y denote the length of an M G 1 busy period, and let F s denote the Laplace-Stieltjes transform of the distribution of y that is, F s E e sFurther, denote the length of service time for the first customer in the busy period by and let denote the number of arrivals during the service time of this customer. Then F s jf E e-S i x...