How and why we calculate the input-output coefficients?

HOW:

- coefficient A = (aij) = Xij/Xj or Xij=aij*Xj; aij –считается только для 1го сектора!

aij- number of units of industry “I” required to produce one unit of industry “j”

aij – говорит о том, какова доля отрасли, которая предоставляет ресурсы в общие ресурсы.

- coefficient M = (mkj) = Zkj/Xj or Zkj=mkj*xj

M – доля издержек в общих ресурсах отрасли

WHY:

We do it for the next analysis.

Construct the distribution equations.

Markov chains

Describe Markov chain as a stochastic process.

• A Markov chain describes a system moving from one state to another under a certain probabilistic rule.

In probability theory and statistics, a Markov process is a stochastic process that satisfies the Markov property.

A Markov chain is a stochastic process on a discrete (finite or countably infinite) space in which the distribution of the next state depends only on the current state. These objects show up in probability and computer science both in discrete-time and continuous-time models. For Markov processes on continuous spaces use markov-process.

What is a transition probability in Markov chain?

Transition probability p_ij – the probability that the process moves from state i at one stage to state j at the next stage.

Compare absorbing and non-absorbing states in the Markov chains.

Absorbing state: the system will stay in the state for ever

Describe the transition matrix with absorbing states.

The transient matrix P is divided into 4 submatrices:

I …transient probabilties between absorbing states (identity matrix)

O… transient probabilities from absorbing to transient states (zeros)

R …transient probabilities from transient to absorbing states

Q …transient probabilities between transient states

What are the steady state probabilities and how can be obtained?

The steady state behavior means that once the process reaches steady state, the state probabilities do not change

The steady state probability of state i is πi

 

Queuing theory

Which are the priority rules for coming from the queue to service?

There are several commonly used rules:

First come first served (FCFS).

=First In First Out (FIFO)

Last come first served (LCFS)

= Last In Last Out (LIFO).

Special Priority (PRI)

Random selection of customers for service. (RND)

Provide an example of tandem queues.

Examples: Patients in an emergency room. Passengers prepare for the next flight.

 

What is a steady state of the queuing system?

In order to achieve steady state, the effective arrival rate must be less than the sum of the effective service rates.Utilization rate: ρ=λ/μ must be <1

Describe the Kendal’s classification of the queuing systems.

Queuing system can be classified by:

• Arrival process.

• Service process.

• Number of servers.

• System size (infinite/finite waiting line).

• Population size

51. Under which necessary conditions the stochastic process can be modelled as Poisson’s? Explain the conditions!

Inventory control

Why the inventory is kept?

- To maintain independence of operations

-To meet variation in product demand

-To allow flexibility in production scheduling

-To provide a safeguard for variation in raw material delivery time

-To take advantage of economic purchase-order size (quantity discount)