Mathematical model of the income change process of the trading companyexpanding the presence in the market

The problem of determining the average income of a trading company and its change duringthe marketing campaign "gift for your purchase» is considered. As mathematical model of numberof companys clients change processes we consider the streams of two-phase queuing systemwith unlimited number of lines and realization of repeated references to phases.The flow of customers for the first time shopping in a trading company, is modeled by theelementary flow of events with parameter ƒ . Having made purchase, the client considers sometime, whether to apply to this company again or choose another. We assume that the duration oftime intervals thinking customers are independent identically distributed random variables witharbitrary distribution function B1( x) . After considering, with probability 1− r1 the client won'taddress any more in the given trading company, preferring it another, and with the probability ofr1 customer returns. At the moment of returning of the client in the company, with probability 1−qpurchases already made by the client in the sum don't exceed some set size. In this case the clientis the client of the first category and the probability of returning in the company at it remains thesame, r1. If the amount of customer purchase in the company exceeds the specified threshold, andit will happen with probability q, then a client becomes a client of the second category, and probabilityof returning in the company and time of considering for it change on r2 and B2(x) respectively.Thus, a flow of repeat clients' appeals, described by random processesn1(t) , n2(t) , wherenk(t) - number of client calls of the k-th category, received by the trading company for the observationtime t. The research problem of total random process n(t) = ƒ(t) +n1(t) +n2(t) in consideredsystem, where ƒ(t ) - is the number of initial references to system, and a finding of itsgenerating function is put.Using the method of limiting decomposition, the form of generating function of the numbersof all clients of the trading company is found. The form of characteristic function of the income ofthe company and its expectation and variance is received. The research of the companys incomechange process under conditions of carrying out of the marketing action «the gift for purchase» ismade. An expression for determining the optimal cost of the gift maximizing the income of thecompany is received. The example of use of the received analytical expressions for calculation ofdesirable cost of a gift, for the purpose of attraction of the maximum number of clients or for theassociated maximizing monetary profit from the action carrying out is resulted.

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