2. UNIT 3. ANALYSIS OF WAITING LINE PROCESSES
Upon completion of the learning unit, the student will be able to understand the concept of
Markov chain. Will be able to understand the operation of waiting line processes and apply
mathematical models to optimize the system. Will be able to understand and apply decision
models as an aid in the evaluation of alternatives. Will be able to apply simulation models as a
tool for future projections and as an aid in decision making.
3. EVIDENCES
EC1: Concept of Markov Processes
EC2: Concept, analysis and operation of waiting line processes.
EC3: Operation of decision models
4. MARKOV CHAIN
Stochastic Processes
In the previous units we have seen how to make models that describe situations corresponding
to the observation of the result of a punctual experiment.
Now we will study the tools to be used when the experiment we want to model is not punctual,
but evolves in time.
For these cases the concept of stochastic process is introduced, defining it as a family of a.v. that
describe the evolution, through time, of a physical process or experiment.
5. We define a stochastic process X=(Xt,∈T) as a family of random variables, whose index t∈T is
called the set of the indices (or time parameter) of process X. For each t ∈ T, Xt is a random
variable over the state space of the process X. We define the state space of a stochastic process X
as the minimal set ∈ of all possible values that the v.a. Xt can assume; each of these values is a
possible state of the system. We note (Xt = e) the event that indicates that at instant t the
process is in state e.
6. If E is a finite or infinite numberable set of states, the process X is said to be a discrete state
space process. If instead the v.a. Xt take their values in a continuous set E (e.g., the positive reals)
the process is said to be of continuous state space.
When T is a numerable set the process (Xt , t ∈ T) is discrete-time (also called stochastic
succession or staged process). The most common case is when T=N (the naturals) or T=N* (the
positive integers). When T is a real interval, finite or not, the process is continuous time.
7. MARKOV CHAIN
Among the different classes of stochastic processes, Markovian processes stand out, in which the
behavior and future evolution of the process depend only on the current state of the process,
and not on its past evolution. Markov chains (M.C.) are discrete state space from a Markovian
processes. As for any process, a distinction is made between discrete-time and continuous-time
Markov chains. The remainder of this section deals with finite discrete-time Markov chains, i.e.
those whose state space E is finite, and which evolve in stages or steps (T is discrete).
8. We formally define a Markov chain as a sequence of discrete a.v.'s {X n EN} that possess the
following property:
(provided that these probabilities are well defined).
This property is interpreted as the statement that the (conditional) probability that the process
reaches the future state xn+1 given the past states x0, x1,...xn-1 and the present state xn is
independent of the past states and depends only on the present state xn (the past does not
influence the future except through the present).
9. CONCEPT, ANALYSIS AND OPERATION OF
WAITING LINE PROCESSES
Arrivals Characteristics:
The service system has three main characteristics:
◦ Arrival Population
◦ Arrival Pattern
◦ Arrivals Behavior
10. SOURCE POPULATION SIZE
Population sizes can be unlimited or limited. An example of an unlimited population could be the
arrival of cars at a booth, customers at a supermarket; since they are a small part of the overall
population. A limited population is one in which your service is small and can be broken down,
such as a stationery store with copiers.
System Arrival Pattern:
Is one in which customers arrive at a service facility; arrivals are considered random, since, they
are independent of each other, and these cannot be accurately predicted. In queuing problems
the arrival numbers can be estimated by a probability distribution known as Poisson distribution.
11. ARRIVALS BEHAVIOR
Customers are either people or machines, waiting their turn to receive a service; unfortunately
people get frustrated or desperate.
Customer defectors are those who enter the line, but become impatient and leave without
completing their transaction.
Queue Line Characteristics
The queue line itself is the second component of a queuing system. A queue is constrained
when it cannot grow to an infinite length; this may be the case for a small barber shop.
A queue is said to be limited when its size is not restricted, such as a toll booth serving motorists.
12. Characteristics of Service Installations
Service systems are classified in terms of their numbers of channels and the number of phases.
◦ Single channel queuing system.
◦ Multi-channel queuing system.
◦ Single phase queuing system.
◦ Multiphase system.
Arrival patterns can be constant or random. If constant, it takes the same amount of time to
service each customer, such as an automatic car wash. More often, service times are random,
and can be assumed to be described by the Negative Exponential Probability Distribution.
19. Model A: Simple channel models with Poisson arrivals and exponential service times.
The conditions for this system are:
◦ Arrivals are serviced on a first-in, first-out (FIFO) basis.
◦ Each arrival is independent of the previous one, but the average number of arrivals does not
change.
◦ Arrivals are described by a Poisson probability distribution and are of Infinite population.
◦ Service times vary from one customer to the next and are independent of each other.
◦ Service times occur according to the negative exponential probability distribution. Service rate
is faster than arrival rate.
20. Model B: Multi-channel queuing model
A multi-channel queuing system in which two or more servers are available to handle arriving
customers.
The multichannel system, presented again assumes that arrivals follow a Poisson probability
distribution and that service times are exponentially distributed.
Service is first-in, first-out and all servers are assumed to perform at the same rate.
21. Model D: Limited population model
When there is a population of potential customers for a service facility, a different queuing model
needs to be considered.
The limited population model allows any number of maintenance people (servers) to be
considered.
The reason this model is different is because there is now a relationship between queue length
and arrival rate.
22. DECISION MODELING PERFORMANCE
WHAT IS DECISION ANALYSIS?
Stephen P. Robbins
How man behaves and acts in order to maximize or optimize a certain outcome, decisions are
made in "reaction" to a problem. There is a discrepancy between the current state of affairs and
the desired state which requires consideration of other courses of action.
23. ELEMENTS THAT CLASSIFY IT
The five most important organizational elements when analyzing a decision have to do with the
following factors
◦ Future effects: This has to do with the extent to which the commitments related to the
decision will affect the future. A decision that has a long-term influence may be considered a
high-level decision, while a decision with short-term effects may be made at a much lower
level.
◦ Reversibility: Refers to the speed with which a decision can be reversed and the difficulty
involved in making this change. If reversing is difficult, it is recommended to take the decision
at a high level; but if reversing is easy, it is required to take the decision at a low level.
24. Impact: This characteristic refers to the extent to which other areas or activities are affected. If
the impact is extensive, it is indicated to take the decision at a high level; a single impact is
associated with a decision taken at a low level.
Quality: This factor refers to labor relations, ethical values, legal considerations, basic principles
of conduct, company image, etc. If many of these factors are involved, a high level of decision
making is required; if only some factors are relevant, a low level of decision making is
recommended.
Periodicity: This element answers the question of whether a decision is made frequently or
exceptionally. An exceptional decision is a high-level decision, while a decision that is made
frequently is a low-level decision.
25.
26.
27. DECISIONS UNDER CONDITIONS OF UNCERTAINTY
◦ Long-term consumer demand for a new good or service.
◦ Exploration cost requirements.
◦ Forecasting technological changes
◦ Forecasting the behavior of stock market values stock market performance.
◦ Forecasting of the general economic state in the medium and medium term and long term, etc.
◦ Machine breakdowns and failures in our own service processes service processes
◦ The distribution of call center waiting times.
◦ The demand for services in a certain geographic area of the Republic.
◦ The fraction of students not passing a certain type of exams, etc. of exams, etc.
28. DECISIONS UNDER CONDITIONS OF CERTAINTY
1. Machine loading problems, i.e., problems with the assignment of different tasks to different
machines, and the scheduling of scheduling of these tasks in the shop.
2. Determination of an optimum product mix.
3. Determination of the optimal production runs, i.e. determination of the i.e. determination of
the number of units of the different of the various products that can be produced
simultaneously.
4. Determination of an optimal transportation plan for the products shipped by products sent by
the factories to the various storage points. Assignment of responsibilities to each crew and
supervisor involved in the work.
