mmap_backward_moment.m

function MOMENTS = mmap_backward_moment(MMAP,ORDERS,NORM)
% Computes the theoretical backward moments of an MMAP.
% Input:
% - MMAP:   the MMAP
% - ORDERS: vector with the orders of the moments to compute
% - NORM:   0 (default) to return B_{i,c}: M_i = sum B_{i,c}
%           1 to return B_{i,c}: M_i = sum B_{i,c} * p_c
%           where
%             M_i is the class independent moment of order i
%             B_{i,c} is the class-c backward moment of order i
%             p_c is the probability of arrivals of class c
% Output:
% - MOMENTS: the backard moments as a matrix B(c,i) = B_{i,c}
 
% by default moments are normalized
if nargin == 2
    NORM = 1; 
end
 
C = length(MMAP)-2;
K = length(ORDERS);
 
if (map_issym(MMAP))
    MOMENTS = sym(zeros(C,K));
else
    MOMENTS = zeros(C,K);
end
 
pie = map_pie(MMAP);
M = inv(-MMAP{1});
 
for a = 1:C
    if NORM
        pa = sum(pie * (-MMAP{1}\MMAP{2+a}));
    else
        pa = 1;
    end
    for h = 1:length(ORDERS)
        k = ORDERS(h);
        fk = factorial(k);
        MOMENTS(a,h) = fk/pa * sum(pie * M^(k+1) * MMAP{2+a});
    end
end
 
end