This is the ninth book of examples from Probability Theory. The topic Stochastic Processes is so big
that I have chosen to split into two books. In the previous (eighth) book was treated examples of
Random Walk and Markov chains, where the latter is dealt with in a fairly large chapter. In this book
we give examples of Poisson processes, Birth and death processes, Queueing theory and other types
of stochastic processes.
Approximation Algorithms for Product-Form Networks
In Chapter 8, several efficient algorithms for the exact solution of queueing networks are introduced. However, the memory requirements and computation time of these algorithms grows exponentially with the number of job classes in the system. For computationally difficult problems of networks with a large number of job classes, we resort to approximation methods. In Sections 9.1, 9.2, and 9.3 we introduce methods for obtaining such approximate results. The first group of methods is based on the MVA.
Algorithms for Non-Product-Form Networks
Although many algorithms are available for solving product-form queueing networks (see Chapters 8 and 9), most practical queueing problems lead to non-product-form networks. If the network is Markovian (or can be Markovized), automated generation and solution of the underlying CTMC via stochastic Petri nets (SPNs) is an option provided the number of states is fewer than a million. Instead of the costly alternative of a discrete-event simulation, approximate solution may be considered.
Algorithms for Product-Form Networks
Although product-form solutions can be expressed very easily as formulae, the computation of state probabilities in a closed queueing network is very time consuming if a straightforward computation of the normalization constant using Eq. (7.3.5) is carried out. As seen in Example 7.7, considerable computation is needed to analyze even a single class network with a small number of jobs, primarily because the formula makes a pass through all the states of the underlying CTMC.
In addition to this paradigm shift from batch and queue to single-piece flow, Lean Manufacturing requires
a systematic elimination of all possible forms of non-value-added costs (e.g., waste). In essence, pollution
is a manifestation of economic waste and is a sign of production inefficiency, revealing flaws in product
design or production processes. It is the unnecessary, inefficient, or incomplete utilization of a resource,
or represents a resource not being used to its highest value.
In its most basic form, Lean Manufacturing is the systematic elimination of waste by focusing on
production costs, product quality and delivery, and worker involvement. In the 1950s, Taiichi Ohno,
developer of the Toyota “just-in-time” Production System, created the modern intellectual and cultural
framework for Lean Manufacturing and waste elimination. Ohno defined waste as “any human activity
which absorbs resources but creates no value.
In this section we will introduce evaluation:
lies in the fact that it can be applied almost unconditionally to all queueing models and at many levels of abstraction. Its strength furthermore lies in the fact that its form is both intuitively appealing and simple. Little’s law and explain it intuitively. A more thorough In Section 2.1.1 we introduce proof is given in
Participants can also be encouraged to select their own anticipated speed for walking the route. This
will help you place the faster walkers at the front of the queue and the more leisurely walkers at the
back, ensuring everyone has an enjoyable experience.
The number of participants released onto the route at one time may need to be managed by stewards
- with participants gathering in a small ‘muster’ area. This control measure allows you to manage the
number of participants starting at any one time and to allow gaps to form if necessary.
A necessary caveat is that we consider a specific LSM, comparing it to a specific model of
internal queues. Other LSMs, perhaps associated with different settlement rules, may yield
different outcomes. For example, one could think of a system where all payments (even those
sent to the RTGS stream) are first passed through the LSM. Then, if LSM settlement does not
happen instantly because a cycle has not formed, the urgent RTGS payments are immediately
settled by transferring liquidity.