. Batch size in systems with general service rule For an M/M

(a, b)/1 system, let w = 1/(A + u), p = 2/u. The distri-

bution of service batch size Y is given by

% = Pr(Y = a)

p Po.o

=1-

[-(r - w) + (1-w)ra+1(a + 1-ar)], w(1-r)2!

gy = Pr(Y = y}

((1-@)

yry pPoo a+1sys b - 1, gb = Pr{Y = b}

b(r -w)

=

p Po.o and wr(1-r)

E[Y] = a 1 + (r - w) p P00 (1 -w)ra+ p Po,0

[1 + r + a - ar]

wr (1-r)2

(1 -r)3

(r - w) p Po,o

"[(1 + 2b) + (1 - 2b)r]}.

@ (1 - r)3 Replacing @ by w, = A/(A + cp), and p by Pc = A/cp, one gets the corresponding expressions for the M/ M

(a, b)/c model (Sim and Templeton, 1985).

Note: This gives the distribution of the size of the batch in which a randomly chosen customer is served. This is to be distinguished from the distribution of the size of a randomly chosen batch.