# Two parallel queues created by arrivals with two demands

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In this thesis we consider the system of two unbounded single server queues in which a customer upon arrival joins both queues. The arrivals are assumed to form a Poisson process with mean interarrival time 1 and the servers have exponential service time distribution with means 1/(alpha) and 1/(beta) respectively. It is assumed 1 < (alpha) (LESSTHEQ) (beta) and hence the equilibrium probabilities p(,ij) are positive for all i,j and.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;The functional equation for the generating function.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;is obtained from the equilibrium equations. This equation exhibits a relation between the functions P(z,0), P(0,w) on the algebraic curve S = {lcub}(z,w): (zw)('2) - (1 + (alpha) + (beta))zw + (alpha)w + (beta)z = 0{rcub}. S is parametrized by a pair of elliptic functions z = z(t), w = w(t) and the functional equation is converted into automorphy conditions for A(t) = P(z(t),0) and B(t) = P(0,w(t)), which are then continued analytically to the whole t-plane. A(t) and B(t), and hence P(z,0), P(0,w), P(z,w) are obtained in closed form. Asymptotic formulas for p(,ij) as i,j (--->) (INFIN) are obtained from the expression for P(z,w).