# 6 service times occur according to the negative

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6.Service times occur according to the negative exponential probability distribution.7.The average service rate is greater than the arrival rate.8.No more than one arrival can occur at a time.3.1 Definition of birth-death process and its relation to queueA birth-death process is a special type of Markov process. As the name implies, birth-death processes were originally used to describe populations that were increased by birthand decreased by death. Biologists, in particular, study the development of animals,plants and other organisms by using birth-death processes. Later, it was found that birth-death processes usually arise when there is a group of entities that are increased by birthor arrivals and decreased by death or departures.A queue, in the sense used here, can be described as a group of elements, such ascustomers waiting for service, machines waiting for repair or cars waiting for at a petrolstation.
Introduction to Waiting Line ManagementMAH2021In the birth-death processes, there is only one state variable,Q, namely, the number ofelements in the population or in the line. Both birth and death rates depend onQ. IfQ=i,the birth rate isiand the death rate is.iSince, a birth increases the population by 1,the rate at whichQincreases fromitoi+1 isi( the probability of increasing thepopulation). Similarly,i(the probability of decreasing the population) is the rate atwhich the populationQdecreases fromitoi1.The states 0, 1, 2, 3, … represent theincrease in population by 0, 1, 2, 3, … respectively.So, the only non-zero transition ratesarei(when the population increases fromitoi+ 1) andi( when the populationdecreases fromitoi1). Transition matrix of the birth-death processes is shown below...3210...3210...........................000...000...000...00003322110Rates always presume the existence of events, such as arrivals or at least changes of state.Consequently, there can be no rate to go from stateito statei. There is, however, a rateof leaving statei. This rate is equal to the sum of the rates of going from statesitoi-1 ori+1, that is,.iiIn the literature, leaving rate of stateiis defined by().iiTheresulting transition matrix is then...3210A =...3210...................................000...0)(.00)(...000332222111100Note that the sum of the probabilities of changing from a state and changing to that stateis equal to zero. If),,...,,,(1210iibe the probability that the system is in statei, then for steady-state probability (probability of any state that remains the same), thesteady state equations will be0A
Introduction to Waiting Line ManagementMAH2021or011000)(22111000)(3322211……………………………………0)(11111iiiiii……………………………………Addition of the 1st2 equations implies that02211Addition of the 1st3 equations implies that03322Similarly, sum of the 1sti

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Term
Spring
Professor
NoProfessor
Tags
Probability theory, Exponential distribution, Poisson process, Queueing theory, Operating Service Facility