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

Calculate, A paper mill produces two grades of paper viz X and Y. Because o...

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 in

Mathematical Model, What is meant by a mathematical model of a real situati...

What is meant by a mathematical model of a real situation? Discuss the importance of models in the solution of Operations research problems

Describe the demand curve, The following equation gives Jane's linear deman...

The following equation gives Jane's linear demand curve for visits to a nearby national park.  Q is the number of visits per week and P represents the price of a pass (price per vi

Experimental method, In this method variables beings studied are contro...

In this method variables beings studied are controlled by the investigator. In other the effect of one variable is observed while other relevant variables are held constant

Survey method, This method involves a systematic and comprehensive study...

This method involves a systematic and comprehensive study of a particular community organization group etc, with a view to the analysis of a social problem and the presen

What do you understand by the term queueing theory, Question: (a) (i) W...

Question: (a) (i) What do you understand by the term ‘queueing theory'? Give two real life examples to illustrate your answer. (ii) Outline the five components involved in

MB0048, A paper mill produces two grades of paper viz., X and Y. Because of...

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

Operation reaserch, 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

Operations research, state phases of operations research and and their impo...

state phases of operations research and and their importance in solving problems

Cost- sampling decisons, Cost The  cost factor  is a very  importan...

Cost The  cost factor  is a very  important  factor in determining  the size of the  sample. The budget for research should be properly  decided. The management may not lik

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