compute probabilities-transitions and transition rate, Operation Research

Assignment Help:

Consider a queueing system with 1 counter, to which groups of customers arrive according to a Poisson proces with intensity l. The size of a group is 1 with probability p and 2 with probability 1-p. Customers are served one by one. The service time has exponential distribution with mean m-1 . Service times are mutually independent and independent of the arrival proces. The system may contain at most 3 customers. If the system is full upon arrival of a group, or if the system may contain only one additional customer upon arrival of a group of size 2, then all customers in the group are lost and will never return. Let Z(t) record the number of customers at time t.

(a)    Explain why {Z(t), t³0}is a Markov proces and give the diagram of transitions and transition rates.

(b)   Give the equilibrium equations (balance equations) for the stationary probabilities Pn , n=0,1,2,3.

(c)    Compute these probabilities Pn , n=0,1,2,3.

The answers to the following questions may be provided in terms of the probabilities Pn  (except for (h)).

(d)   Give an expression for the average number of waiting customers.

(e)   Give the departure rate and the rate at which customers enter the queue.

(f)    Give an expression for the average waiting time of a customer.

(g)   What is the fraction of time the counter is busy?

(h)   What is the average length of an idle period?

(i)    Determine from  (g) and (h) the average length of a period the system is occupied (= at least 1 customer in the system).

(j)    What is the rate at which groups of size 2 enter the queue?


Related Discussions:- compute probabilities-transitions and transition rate

HOMEWORK, A company uses 4 pounds of resource 1 to make each unit of X1 and...

A company uses 4 pounds of resource 1 to make each unit of X1 and 3 pounds of resource 1 to make each unit of X2. There are only 150 pounds of resource 1 available. Which of the fo

Uses of mode - measure of central tendency, Uses of Mode The  use of t...

Uses of Mode The  use of the mode is recommended in the following  situations: a.When a quick  and approximate measures of  central tendency desired. b.When the  measure

the simplex method, In large sized linear programming problems, the ...

In large sized linear programming problems, the solution cannot be obtained by the graphical method and hence a more systematic method has to be developed to find

OPERATION RESEARCH, A paper mill produces two grades of paper viz., X and Y...

A paper mill produces two grades of paper viz., X and Y. Because of raw material restrictions, it cannot produce more than 400 tons of grade X paper and 300 tons of grade Y paper i

Decision trees, Ask questioA farmer intends to drill a well for his herd. I...

Ask questioA farmer intends to drill a well for his herd. In the past, only 80% of the wells drilled in this area were successful at 12 metres of depth. On finding no water at 12 m

Game Theory, Task1:- A company is currently involved in negotiation with it...

Task1:- A company is currently involved in negotiation with its union on the upcoming wage contract. Positive signs in the table represent wage increase while negative sign represe

Disadvantage of standard deviation - measure of dispersion, Disadvantage ...

Disadvantage  of standard Deviation It is not  easy  to understand  and calculate. It gives  more  weight  to the extremes  and less  to the items   nearer to mean. Since

Considerations for oral reporting, Considerations for Oral Reporting S...

Considerations for Oral Reporting Sometimes  the researcher has been asked to  make an oral presentation of his  research findings he should  bear in mind a few major  conside

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd