Appendix I
1.1 Proof of Theorem 3.2.1
By substituting \( {\theta_1}={\lambda_2}+\alpha +{\lambda_1}\left( {1-zk!} \right)-1, \) the LHS of Eq. (13) vanishes and we have
$$ \begin{array}{*{20}c} {Q_{k,h}^{*}\left( {0;z} \right)\left( {1-{\lambda_1}zS^{*}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-zk!} \right)-1} \right)-\frac{2}{z}} \right)=\frac{{\alpha \left( {k+1} \right)!}}{z}Q_{k+1,h}^{*}\left( {0;z} \right)+} \hfill \\ {\beta \left( {k-1} \right)!P_{k-1,h}^{*}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-zk!} \right)-1;z} \right)+{\lambda_1}{P_{k,h }}(z)} \hfill \\ {h=1} \hfill \\ \end{array} $$
(A1)
Substituting this in Eq. (12) and simplifying, we get
$$ {P_{k,h }}(z)=\frac{{\left( {\alpha \left( {k+1} \right)!Q_{k+1,h}^{*}\left( {0;z} \right)+z\ \beta \left( {k-1} \right)!P_{k-1,h}^{*}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-zk!} \right)-1;z} \right)+z\beta {P_{k-1,h }}\left( {0;z} \right)} \right){S^{*}}\left( {{\theta_1}} \right)}}{{\left( {{\lambda_1}+{\lambda_2}+\alpha } \right)\left( {z-{\lambda_1}{z^2}{S^{*}}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-zk!} \right)-1} \right)-2} \right)-{\lambda_1}z{S^{*}}\left( {{\theta_1}} \right)}} $$
(A2)
1.2 Proof of Theorem 3.2.2
By substituting \( {\nu_1}={\lambda_1}+\alpha +{\lambda_2}\left( {1-wk!-\left( {k-1} \right)!} \right)-1, \) the LHS of Eq. (17) vanishes and we have
$$ \begin{array}{*{20}c} {Q_{k,h}^{*}\left( {0;w} \right)\left( {1-{\lambda_2}wS^{*}\left( {{\lambda_1}+\alpha +{\lambda_2}\left( {1-wk!-\left( {k-1} \right)!} \right)-1} \right)-\frac{2}{w}} \right)=\frac{{\alpha \left( {k+1} \right)!}}{w}Q_{k+1,h}^{*}\left( {0;w} \right)+} \hfill \\ {\beta \left( {k-1} \right)!P_{k-1,h}^{*}\left( {{\lambda_1}+\alpha +{\lambda_2}\left( {1-wk!} \right)-1;w} \right)+{\lambda_2}{P_{k,h }}(w)\quad \quad h=2} \hfill \\ \end{array} $$
(A3)
Substituting this in Eq. (16) and simplifying, we get
$$ {P_{k,h }}(w)=\frac{{\left( {\alpha \left( {k+1} \right)!Q_{k+1,h}^{*}\left( {0;w} \right)+w\ \beta \left( {k-1} \right)!P_{k-1,h}^{*}\left( {{\lambda_2}\left( {1-wk!} \right)+\alpha +{\lambda_1}-1;w} \right)+w\beta {P_{k-1,h }}\left( {0;w} \right)} \right){S^{*}}\left( {{\nu_1}} \right)}}{{\left( {{\lambda_1}+{\lambda_2}+\alpha } \right)\left( {w-{\lambda_2}{w^2}{S^{*}}\left( {\left( {{\lambda_2}} \right.\left( {1-wk!} \right.-\left( {k-1} \right)!} \right)+\alpha +\left. {{\lambda_1}-1} \right)-2} \right)-{\lambda_2}w{S^{*}}\left( {{v_1}} \right)}} $$
(A4)
1.3 Proof of Theorem 3.2.3
By substituting \( {\theta_i}={\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! } \) for i = 1,2…,n, the LHS of Eq. (14) vanishes and we have
$$ \begin{array}{*{20}c} \begin{array}{*{20}c} {Q_{k,h}^{*}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! },\ldots,{\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! };z} \right)=} \hfill \\ {\frac{{\left( {\begin{array}{*{20}c} {\frac{{\alpha \left( {k+1} \right)!}}{zn! }Q_{k+1,h}^{*}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! },\ldots,{\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! };z} \right)+} \hfill \\ {\frac{{\beta \left( {k-1} \right)!}}{n! }P_{k-1,h}^{*}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! },\ldots,{\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! };z} \right)} \hfill \\ \end{array}} \right)}}{{1-\frac{{{\lambda_1}z}}{n}S^{*}\left( {{\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! }} \right)-\frac{{\left( {n+1} \right)}}{z}}}} \hfill \\ {2\leq k\leq C-1\quad, \quad 1\leq i\leq n,h=1} \hfill \\ \end{array} \hfill \\ \kern28.5em \hfill \\\end{array} $$
(A5)
which is the another expression of \( P_{k,h}^{*}\left( {{\theta_1},{\theta_2},\ldots,{\theta_n};z} \right),\;h=1, \) where \( {\theta_i}={\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! } \) in terms of \( Q_{k,h}^{*}\left( {{\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n};z} \right) \), for type 1 customers.
Differentiating both sides of (13), (r + 1) times with respect to θ
1 and evaluating at \( {\theta_1}={\lambda_2}+\alpha +{\lambda_1}\left( {1-zk!} \right)-1 \) yields for r = 0, 1, 2…,:
$$ \left( {r+1} \right)P_{k,h}^{*(r)}\left( {{\theta_1};z} \right)=-{\lambda_1}z{S^{{*\left( {(r+1} \right)}}}\left( {{\theta_1}} \right)Q_{k,h}^{*}\left( {0;z} \right)-\beta \left( {k-1} \right)!P_{k-1,h}^{{*\left( {r+1} \right)}}\left( {{\theta_1};z} \right),\;h=1,n=1 $$
(A6)
By adding equations (13) and (14), and differentiating both sides, r + 1 times w.r.t. θ
i
for i = 1,2,…,n ≤ k, and evaluating at \( {\theta_i}={\lambda_2}+\alpha +{\lambda_1}\left( {1-\frac{{z\left( {k-n+1} \right)!}}{n! }} \right)-\frac{1}{n! } \) yields for r = 0, 1, 2…,:
$$ \begin{array}{*{20}c} {\left( {r + 1} \right)P_{{k,\;h}}^{*(r)}\left( {{\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;z} \right) = -\sum\limits_{{i = 1}}^n {\frac{{{\lambda_1}z}}{n}{S^{{*\left( {()} \right)}}}\left( {{\theta_i}} \right)} Q_{{k,\,h}}^{*}\left( {{\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}}\ldots ..{\theta_n};\;z} \right) + } \hfill \\ {\sum\limits_{{i = 1}}^n {\left( {1 - \frac{{{\lambda_1}z}}{n}S*\left( {{\theta_i}} \right) - \frac{{\left( {n + 1} \right)}}{z}} \right)Q_{{k,\;h}}^{{*\left( {r + 1} \right)}}\left( {{\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}}\ldots ..{\theta_n};\;z} \right)} } \hfill \\ { - \frac{{\alpha \left( {k + 1} \right)!}}{n!z }Q_{{k + 1,\;h}}^{{*\left( {r + 1} \right)}}\left( {{\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n};\;z} \right) - \frac{{\beta (k - 1)!}}{n! }P_{{k - 1,\;h}}^{{*(r + 1)}}\left( {{\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;z} \right),\;h = 1} \hfill \\ \end{array} $$
(A7)
By starting with r = 0 in the Eq. (A7) and successive substitution of the Eq. (A7), with r = 1, 2,…,k we get
$$ \begin{array}{*{20}c} {P_{k,h}^{*}\left( {{\theta_1},{\theta_2},\ldots,{\theta_n};z} \right)=P_{k,h}^{*(0)}\left( {{\theta_1},{\theta_2},\ldots,{\theta_n};z} \right)=} \hfill \\ {\sum\limits_{l=1}^k {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{n\;!(1.2\ldots (k+1-l))}}} \sum\limits_{i=1}^n {\frac{{{\lambda_1}z}}{n}{S^{*(k+1-l) }}\left( {{\theta_i}} \right)} Q_{l,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n}; z\;)-} \hfill \\ {\ \sum\limits_{l=1}^k {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{n\;!(1.2\ldots (k+1-l))}}} \sum\limits_{i=1}^n {\left( {1-\frac{{{\lambda_1}z}}{n}{S^{*}}\left( {{\theta_i}} \right)-\frac{(n+1) }{z}} \right)Q_{l,h}^{*(r+1) }({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n}; z\;)} } \hfill \\ { - \sum\limits_{l=1}^{k+1 } {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{n\;!(1.2\ldots (k+1-l))}}} \sum\limits_{i=1}^n {\frac{{l\;!\alpha }}{{z\;n\;!}}Q_{l,h}^{*(k+1-l }({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n}; z\;)} \kern0.75em ,\ h=1,\ 1\leq k\leq C} \hfill \\ \end{array} $$
(A8)
By substituting (A8) in (A5), we have the following recursive form about the boundary function
$$ \begin{array}{*{20}c} {Q_{k,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n}; z\;),\kern1em h=1} \hfill \\ \begin{array}{*{20}c} Q_{k,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n}; z\;)=\left\{ \begin{array}{*{20}c} \frac{{\frac{{\alpha 2\;!}}{{z\;}}Q_{2,h}^{*}(0;z\;)-{\lambda_1}{P_{1,h }}(z)}}{{1-{\lambda_1}zS*\left( {{\theta_i}} \right)-\frac{2}{z}}}\kern0.75em ,\ k=1,h=1\kern1em \hfill \\ \frac{{\left( \begin{array}{*{20}c} \frac{{\alpha (k+1)\;!}}{{z\;n\;!}}Q_{k+1,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n}; z\;) + \hfill \\ \frac{{\beta (k-1)\;!}}{{n\;!}}P_{k-1,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_n}; z\;) \hfill \\\end{array} \right)}}{{1-\frac{{{\lambda_1}z}}{n}S^{*}\left( {{\theta_i}} \right)-\frac{(n+1) }{z}}}\kern0.5em ,\ 2\leq k\leq C-1,\ 1\leq i\leq n,\,h=1 \hfill \\ \frac{{\frac{{\beta (C-1)\;!}}{{C\;!}}P_{C-1,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_C}; z\;)}}{{1-\frac{{{\lambda_1}z}}{C}S^{*}\left( {{\theta_C}} \right)}}\kern0.75em ,\ k=C,h=1 \hfill \\\end{array} \right.\ \hfill \\ \kern20.75em \hfill \\\end{array} \hfill \\ \end{array} $$
From the above equations, we get
$$ \begin{array}{*{20}c} {Q_{2,h}^{*}(0;z) = \frac{\mathrm{z}}{{\alpha 2\;!}}\left( {Q_{1,h}^{*}(0;z)\left( {1 - {\lambda_1}zS^{*}\,\left( {{\theta_1}} \right) - \frac{2}{z}} \right) - {\lambda_1}{P_{{1,\;h}}}(z)} \right),\;\;\;k = 1,\;h = 1} \hfill \\ \begin{array}{*{20}c} Q_{{k + 1,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n};z) = \frac{{z\;n\;!}}{{\alpha (k + 1)\;!}}\left[ \begin{array}{*{20}c} Q_{{k,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n};\;z)\left( {1 - \frac{{{\lambda_1}z}}{n}S^{*}\,\left( {{\theta_i}} \right) - \frac{{(n + 1)}}{z}} \right)- \hfill \\ \kern7em \frac{{\beta (k - 1)\;!}}{{n\;!}}P_{{k - 1,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;z) \hfill \\\end{array} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2\,\leq\,k\,\leq\,C - 1,\;1\,\leq\,i\,\leq\,n,\;h = 1 \hfill \\ Q_{{k,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n};\;z) = \frac{{\frac{{\beta (C - 1)\;!}}{{C\;!}}P_{{C - 1,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_C};\;z)}}{{1 - \frac{{{\lambda_1}z}}{C}S^{*}\left( {{\theta_C}} \right)}},\;\ k = C,\;h = 1 \hfill \\\end{array} \hfill \\ {} \hfill \\ \end{array} $$
(A9)
where \( {\theta_i} = {\lambda_2} + \alpha + {\lambda_1}\left( {1 - \frac{{z(k - n + 1)!}}{\mathrm{n}!}} \right) - \frac{1}{n! } \) and \( P_{{k - 1,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;z),\;h = 1 \) is obtained from (A8).
1.4 Proof of Theorem 3.2.4
By substituting \( {\nu_j} = {\lambda_2}\left( {1 - \frac{{(k - 1)!}}{\mathrm{m}!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_1} - \frac{1}{m! } \) , the LHS of equation (18) vanishes for j = 1,2,…,m, and we have
$$ \begin{array}{*{20}c} Q_{{k,\;h}}^{*}\left( {{\lambda_2}\left( {1 - \frac{{(k - 1)!}}{m!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_1} - \frac{1}{m! },\ldots,\;{\lambda_2}\left( {1 - \frac{{(k - 1)!}}{m!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_1} - \frac{1}{m! };\;w} \right)= \hfill \\ \kern8.25em \frac{{\left( \begin{array}{*{20}c} \frac{{\alpha (k + 1)\;!}}{{wm\;!}}Q_{{2,\;h}}^{*}\left( \begin{array}{*{20}c} {\lambda_2}\left( {1 - \frac{{(k - 1)!}}{m!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_1} - \frac{1}{m! },\ldots, \hfill \\ {\lambda_2}\left( {1 - \frac{{(k - 1)!}}{m!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_1} - \frac{1}{m! };w \hfill \\\end{array} \right) + \hfill \\ \frac{{\beta (k - 1)\;!}}{{m\;!}}P_{{k - 1,\;h}}^{*}\left( \begin{array}{*{20}c} {\lambda_2}\left( {1 - \frac{{(k - 1)!}}{m!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_{{1\,}}} - \frac{1}{m! },\ldots, \hfill \\ {\lambda_2}\left( {1 - \frac{{(k - 1)!}}{m!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_1} - \frac{1}{m! };w \hfill \\\end{array} \right) \hfill \\\end{array} \right)}}{{1 - \frac{{{\lambda_2}w}}{m}S*\left( {{\lambda_2}\left( {1 - \frac{{(k - 1)!}}{m!} - \frac{{w(k - m + 1)!}}{\mathrm{m}!}} \right) + \alpha + {\lambda_1} - \frac{1}{m! }} \right) - \frac{{(m + 1)}}{w}}} \hfill \\ \kern28.75em 2\,\leq\,k\,\leq\,C - 1,\;1\,\leq\,j\,\leq\,m,\;h = 2 \hfill \\\end{array} $$
(A10)
which is the another expression of \( P_{{k,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;w),\;\;\;\;\;h = 2 \) where \( {\nu_j} = {\lambda_1} + \alpha + {\lambda_2}\left( {1 - \frac{{z(k - n + 1)!}}{\mathrm{n}!} - \frac{{(k - 1)!}}{m! }} \right) - \frac{1}{m! } \) in terms of \( Q_{{k,\,h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};w) \), for type 2 customers.
By differentiating both sides of (17), \( (r+1) \) times w.r.t. \( {\nu_1} \), and evaluating at \( {\nu_1} = {\lambda_1} + \alpha + {\lambda_2}\left( {1 - wk! - (k - 1)!} \right) - 1 \) yields for \( r = 0,\;1,\;2.... \), we have
$$ (r + 1)P_{{k,\;h}}^{*(r) }({\nu_1};\;w) = - {\lambda_2}w{S^{{*(r + 1)}}}({\nu_1})Q_{{k,\;h}}^{*}(0;\;w) - \beta (k - 1)\;!P_{{k - 1,\;h}}^{{*(r + 1)}}({\nu_1};\;w),\;h = 1,\,m = 1 $$
(A11)
By adding the equations (17) and (18), and differentiating both sides , \( (r + 1) \) times w.r.t. \( {\nu_j} \) for \( j = 1,\;2,\ldots,\;m\,\leq\,k \), and evaluating at \( {\nu_j} = {\lambda_1} + \alpha + {\lambda_2}\left( {1 - \frac{{z(k - n + 1)!}}{\mathrm{n}!} - \frac{{(k - 1)!}}{m! }} \right) - \frac{1}{m! } \) yields for \( r = 0,\;1,\;2.... \),:
$$ \begin{array}{*{20}c} {(r + 1)\left( {1 + \frac{{(k - 1)!}}{m! }} \right)P_{{k,\,h}}^{*(r) }({\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;w) = - \sum\limits_{{j = 1}}^m {\frac{{{\lambda_2}w}}{m}{S^{{*(r + 1)}}}({\nu_j})} Q_{{k,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}}.....{\nu_m};w) + } \\ {\sum\limits_{{j = 1}}^m {\left( {1 - \frac{{{\lambda_2}w}}{m}S*({\nu_j}) - \frac{{(m + 1)}}{w}} \right)Q_{{k,\;h}}^{{*(r + 1)}}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;w)} } \\ { - \frac{{\alpha (k + 1)!}}{wm! }Q_{{k + 1,\;h}}^{{*(r + 1)}}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\,{\nu_m};\;w) - \frac{{\beta (k - 1)!}}{m! }P_{{k - 1,\;h}}^{{*(r + 1)}}({\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;w),\;h = 2} \\ \end{array} $$
(A12)
By starting with r = 0 in the equation (A12) and successive substitution of the equation (A12), with \( r = 1,\;2,\ldots,\;k \) we get
$$ \begin{array}{*{20}c} \begin{array}{*{20}c} {P_{{k,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;w) = P_{{k,\;h}}^{*(0) }({\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;w) = } \hfill \\ {\sum\limits_{{l = 1}}^k {\tfrac{{{(-1)^{{k + 1 - l}}}{\beta^{{k - 1}}}}}{{m!(1.2\ldots (k + 1 - l))}}} \sum\limits_{{j = 1}}^m {\frac{{\frac{{{\lambda_2}w}}{m}{S^{{*(k + 1 - l)}}}\left( {{\nu_j}} \right)}}{{\left( {1 + \frac{{(k - 1)!}}{m! }} \right)}}} Q_{{l,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,{\nu_{{i - 1}}},\;{\nu_{{i + 1}}},\ldots,\;{\nu_m};\;w) - } \hfill \\ {\ \sum\limits_{{l = 1}}^k {\tfrac{{{(-1)^{{k + 1 - l}}}{\beta^{{k - 1}}}}}{{m!(1.2\ldots (k + 1 - l))}}} \sum\limits_{{j = 1}}^m {\frac{{\left( {1 - \frac{{{\lambda_2}w}}{m}{S^{*}}\left( {{\nu_j}} \right) - \frac{{(m + 1)}}{w}} \right)}}{{\left( {1 + \frac{{(k - 1)!}}{m! }} \right)}}Q_{{l,\,h}}^{{*(r + 1)}}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;w)} } \hfill \\ { - \sum\limits_{{l = 1}}^{{k + 1}} {\tfrac{{{(-1)^{{k + 1 - l}}}{\beta^{{k - 1}}}}}{{m!(1.2\ldots (k + 1 - l))}}} \sum\limits_{{j = 1}}^m {\frac{{\left( {\frac{{l!\alpha }}{{w\;m!}}} \right)}}{{\left( {1 + \frac{{(k - 1)!}}{m! }} \right)}}Q_{{l,\,h}}^{{*(k + 1 - l)}}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;w)},\;h = 2,\;1\,\leq\,k\,\leq\,C} \hfill \\ \end{array} \hfill \\ \kern2.25em \hfill \\\end{array} $$
(A13)
By substituting (A13) in (A10), we have the following recursive form about the boundary function
$$ \begin{array}{*{20}c} {Q_{{k,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;w),\;\mathrm{h} = 2} \hfill \\ {Q_{{k,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_n};\;z) = \left\{ \begin{array}{*{20}c} \frac{{\frac{{\alpha 2!}}{w}Q_{{2,\;h}}^{*}(0;\;w) - {\lambda_2}{P_{{1,\;h}}}(w)}}{{1 - {\lambda_2}zS*\left( {{\nu_j}} \right) - \frac{2}{w}}}\kern0.5em ,\;\;\;\;\;k=1,h=2\kern1em \hfill \\ \frac{{\left( \begin{array}{*{20}c} \frac{{\alpha (k + 1)!}}{{w\;m!}}Q_{{k + 1,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\,{\nu_m};\;w) + \hfill \\ \frac{{\beta (k - 1)!}}{m! }P_{{k - 1,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;w) \hfill \\\end{array} \right)}}{{1 - \frac{{{\lambda_2}w}}{m}S*\left( {{\nu_j}} \right) - \frac{{(m + 1)}}{w}}}\kern0.5em ,\;\ 2\,\leq\,k\,\leq\,C - 1,\;1\,\leq\,j\,\leq\,m,\;h = 2 \hfill \\ \frac{{\frac{{\beta (C - 1)!}}{C! }P_{{C - 1,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_C};\;w)}}{{1 - \frac{{{\lambda_2}w}}{C}S*\left( {{\nu_C}} \right)}},\ \hfill \\ \kern24em k = C,\;h = 2 \hfill \\\end{array} \right.\kern0.75em } \hfill \\ \end{array} $$
From the above equations, we get
$$ \begin{array}{*{20}c} {Q_{{2,\;h}}^{*}(0;\;w) = \frac{\mathrm{w}}{{\alpha 2!}}\left( {Q_{{1,\;h}}^{*}(0;\;w)\left( {1 - {\lambda_2}wS*\left( {{\nu_1}} \right) - \frac{2}{w}} \right) - {\lambda_2}{P_{{1,\;h}}}(w)} \right),\;\;\mathrm{k} = 1,\;\;\mathrm{h} = 2} \hfill \\ \begin{array}{*{20}c} Q_{{k + 1,\;h}}^{*}\left( {{\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;w} \right) = \frac{{w\,m!}}{{\alpha \left( {k + 1} \right)!}}\left[ \begin{array}{*{20}c} Q_{{k,\;h}}^{*}\left( {{\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;w} \right)\left( {1 - \frac{{{\lambda_2}w}}{m}S\,*\,\left( {{\nu_j}} \right) - \frac{{(m + 1)}}{w}} \right) - \hfill \\ \kern7em \frac{{\beta (k - 1)!}}{m! }P_{{k - 1,\;h}}^{*}({\nu_1},\;{\nu_2}.........{\nu_m};\;w) \hfill \\\end{array} \right] \hfill \\ \kern23.25em 2\,\leq\,k\,\leq\,C - 1,\;1\,\leq\,j\,\leq\,m,\;h = 2 \hfill \\\end{array} \hfill \\ \begin{array}{*{20}c} Q_{{k,\,h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\,{\nu_m};\;w) = \frac{{\frac{{\beta (C - 1)!}}{C! }P_{{C - 1,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_C};\;w)}}{{1 - \frac{{{\lambda_2}w}}{C}S*\left( {{\theta_C}} \right)}}\kern0.75em \hfill \\ \kern23.5em k = C,\;h = 2 \hfill \\\end{array} \hfill \\ \end{array} $$
(A14)
where \( {\nu_j} = {\lambda_1} + \alpha + {\lambda_2}\left( {1 - \frac{{z(k - n + 1)!}}{\mathrm{n}!} - \frac{{(k - 1)!}}{m! }} \right) - \frac{1}{m!}\;\mathrm{and}\;P_{{k - 1,\;h}}^{*}({\nu_1},\;{\nu_2},\ldots,\;{\nu_n};\;w),\;h = 2 \) is obtained from (A13).
1.5 Proof of Theorem 3.2.5
The LHS of equation (21) vanishes at \( \left( {{\theta_i}+{\nu_j}} \right)={\lambda_1}\left( {1-\frac{{z(k-n-m+1)\;!}}{{m\;!n\;!}}} \right)+{\lambda_2}\left( {1-\frac{{w(k-n-m+1)\;!}}{{n\;!m\;!}}} \right)+\alpha \)
For \( i=1,2,\ldots,n\;\;\quad j=1,2,\ldots,m \), thus we have
$$ \begin{array}{*{20}c} {Q_{{k,\;h}}^{*}({\theta_1},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n},\;{\nu_1},\ldots,\;{\nu_m};\;z;\;w) + Q_{{k,\;h}}^{*}({\theta_1},\ldots,\;{\theta_n},\;{\nu_1},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;z;\;w) = } \hfill \\ {\ \frac{{\left( \begin{array}{*{20}c} \frac{{\alpha (\mathrm{k} + 1)!}}{\mathrm{n}!\mathrm{z}}\left( {Q_{{k + 1,\;h}}^{*}({\theta_1},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n},\;{\nu_1},\ldots,\;{\nu_m};\;z;\;w) + Q_{{k + 1,\;h}}^{*}({\theta_1},\ldots,\;{\theta_n},\;{\nu_1},, ..,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;z;\;w)} \right) \hfill \\ - \frac{{\beta (\mathrm{k} - 1)!}}{\mathrm{m}!\mathrm{w}}P_{{k - 1,\;h}}^{*}({\theta_1},\ldots,\;{\theta_n},\;{\nu_1},\ldots,\;{\nu_m};\;z;\;w) + \hfill \\ \sum\limits_{{i = 1}}^n {\sum\limits_{{j = 1}}^{{m + i}} {\frac{{{\lambda_2}}}{m! }S*\left( {{\nu_j}} \right)Q_{{k,\;h}}^{*}({\theta_1},\;{\theta_2}....{\theta_{{j - 1}}},\;{\nu_j},\;{\theta_{{j + 1}}}.....{\theta_n};\;{\nu_1},\;{\nu_2}.....{\nu_{{j - 1}}},\;{\nu_{{j + 1}}}....{\nu_m};\;z;\;w)} } \hfill \\\end{array} \right)}}{{\left( {1 - \frac{{{\lambda_1}z}}{n}{S^{*}}\left( {{\theta_i}} \right) - \frac{{n + 1}}{z}} \right) + \left( {1 - \frac{{{\lambda_2}w}}{m}{S^{*}}\left( {{\nu_j}} \right) - \frac{{m + 1}}{w}} \right)}},} \hfill \\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;h = 1,\;2} \hfill \\ \end{array} $$
(A15)
where
$$ \begin{array}{*{20}c} {{\theta_{\mathrm{i}}}={\lambda_1}\frac{{z(k-n-m+1)\;!}}{{m\;!n\;!}}-{\lambda_2}\frac{{w(k-n-m+1)\;!}}{{n\;!m\;!}}+\frac{{{\lambda_2}(k-1)!}}{m! }-\frac{1}{m! },\kern1.25em } \hfill \\ {{\nu_{\mathrm{j}}}=-{\lambda_1}\frac{{z(k-n-m+1)\;!}}{{m\;!n\;!}}+{\lambda_1}\frac{{z(k-n+1)\;!}}{{n\;!m\;!}}-\frac{1}{n! },\kern0.5em } \hfill \\ \end{array} $$
is the another expression for \( P_{k,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_n}; {\nu_1},{\nu_2},\ldots, {\nu_m}; z; w) \) in terms of \( Q_{k,h}^{*}({\theta_1},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+Q_{k,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z;w) \) for the combinations both the customers.
By differentiating both sides of (20), \( (r+1) \) times w. r. t. \( {\theta_1} \) and \( {\nu_1} \) and evaluating at \( {\theta_1}={\lambda_1}z(k-1)\;! - {\lambda_2}w(k-1)\;!+{\lambda_2}(k-1)!-1,\kern1.25em {\nu_1}=-{\lambda_1}z(k-1)\;!+{\lambda_1}z(k-1)\;!-1,\kern1em \)yields for \( r=0,1,2...., \) we have
$$ \begin{array}{*{20}c} {(r + 1)\left( {\frac{{{d^r}}}{{d\theta_1^r}} + \frac{{{d^r}}}{{d\nu_1^r}}} \right)P_{{k,\;t}}^{*}({\theta_1};\;{\nu_1};\;z;\;w) = - > {\lambda_1}z{S^{{*(r + 1)}}}\left( {{\theta_1}} \right)Q_{{k,\;t}}^{*}(0;\;{\nu_1};\;z;\;w) + } \hfill \\ {\left( {1 - {\lambda_1}zS*\left( {{\theta_1}} \right) - \frac{2}{z}} \right)Q_{{k,\;t}}^{{*(r + 1)}}(0;\;{\nu_1};\;z;\;w) - \frac{{\alpha (k + 1)!}}{\mathrm{z}}Q_{{k + 1,\,t}}^{{*(r + 1)}}(0;\;{\nu_1};\;z;\;w) - } \hfill \\ {{\lambda_2}w{S^{{*(r + 1)}}}\left( {{\nu_1}} \right)Q_{{k,\,t}}^{*}({\theta_1};\;0;\;z;\;w) + \left( {1 - {\lambda_2}wS*\left( {{\nu_1}} \right) - \frac{2}{w}} \right)Q_{{k,\;t}}^{{*(r + 1)}}({\theta_1};\;0;\;z;\;w)} \hfill \\ \begin{array}{*{20}c} - \frac{{\alpha (k + 1)!}}{\mathrm{w}}\sum\limits_{{j = 1}}^m {Q_{{k + 1,\;t}}^{{*(r + 1)}}} ({\theta_1};\;0;\;z;\;w) - \beta (k - 1)!\left( {\frac{{{d^r}}}{{d\theta_1^r}} + \frac{{{d^r}}}{{d\nu_1^r}}} \right)P_{{k - 1,\;t}}^{*}({\theta_1};\;{\nu_1};\;z;\;w) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;h = 1,\;2\ \hfill \\\end{array} \hfill \\ \end{array} $$
(A16)
Now, adding equations (20) and (21), and differentiating both sides , \( (r+1) \) times w.r.t \( {\theta_i} \) and \( {\nu_j} \) for \( i=1,2,\ldots n\leq k,\;\;j=1,2,\ldots,m\leq k, \)
m + n
\( \leq \) C and evaluating at
$$ \begin{array}{*{20}c} \begin{array}{*{20}c} {{\theta_{\mathrm{i}}} = {\lambda_1}\frac{{z(k - n - m + 1)!}}{{m\;!n\;!}} - {\lambda_2}\frac{{w(k - n - m + 1)\;!}}{{n\;!m\;!}} + \frac{{{\lambda_2}(k - 1)!}}{m!} - \frac{1}{m! },} \hfill \\ {{\nu_{\mathrm{j}}}=-{\lambda_1}\frac{{z(k - n - m + 1)\;!}}{{m\;!n\;!}} + {\lambda_1}\frac{{z(k - n + 1)\;!}}{{n\;!m\;!}} - \frac{1}{n! },} \hfill \\ \end{array}\kern1.5em \hfill \\ \kern1.25em \hfill \\\end{array} $$
yields for \( r=0,1,2...., \) we have
$$ \begin{array}{*{20}c} \begin{array}{*{20}c} {(r + 1)\left( {\frac{{{\partial^r}}}{{\partial \theta_i^r}} + \frac{{{\partial^r}}}{{\partial \nu_j^r}}} \right)P_{{k,\,h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;z;\;w) = } \hfill \\ {-\sum\limits_{{i = 1}}^n {\frac{{{\lambda_1}z}}{n}\frac{{{\partial^{{r + 1}}}}}{{\partial \theta_i^{{r + 1}}}}{S^{*}}\left( {{\theta_i}} \right)Q_{{k,\,h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,{\theta_n}; {\nu_1},{\nu_2},\ldots,{\nu_m}; z; w\;)} -} \hfill \\ {\sum\limits_{{j = 1}}^m {\frac{{{\lambda_2}w}}{m}\frac{{{\partial^{{r + 1}}}}}{{\partial \nu_j^{{r + 1}}}}{S^{*}}\left( {{\nu_j}} \right)} Q_{{k,\,h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;z;\;w) + } \hfill \\ {\ \sum\limits_{{i = 1}}^n {\left( {1 - \frac{{{\lambda_1}z}}{n}S*\left( {{\theta_i}} \right) - \frac{{(n + 1)}}{z}} \right)\frac{{{\partial^{{r + 1}}}}}{{\partial \theta_i^{{r + 1}}}}Q_{{k,\,h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;z;\;w)} + } \hfill \\ {\sum\limits_{{j = 1}}^m {\left( {1 - \frac{{{\lambda_2}w}}{m}S*\left( {{\nu_j}} \right) - \frac{{(m + 1)}}{w}} \right)\frac{{{\partial^{{r + 1}}}}}{{\partial \nu_j^{{r + 1}}}}Q_{{k,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\,z;\;w)} - } \hfill \\ {\frac{{\alpha (k + 1)!}}{{n\;!\mathrm{z}}}\frac{{{\partial^{{r + 1}}}}}{{\partial \theta_i^{{r + 1}}}}Q_{{k + 1,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\theta_{{i + 1}}},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;z;\;w) - } \hfill \\ {\frac{{\alpha (k + 1)!}}{m!w}\frac{{{\partial^{{r + 1}}}}}{{\partial \nu_j^{{r + 1}}}}Q_{{k + 1,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\,z;\;w) - } \hfill \\ {\frac{{\beta (k - 1)!}}{m!n!}\left( {\frac{{{\partial^r}}}{{\partial \theta_i^r}} + \frac{{{\partial^r}}}{{\partial \nu_j^r}}} \right)P_{{k - 1,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;z;\;w) - } \hfill \\ {\ \sum\limits_{{\mathrm{i} = 1}}^{\mathrm{n}} {\sum\limits_{{\mathrm{j} = 1}}^{{\mathrm{m} + \mathrm{i}}} {\frac{{{\lambda_2}}}{\mathrm{m}!}} } \frac{{{\partial^{{r + 1}}}}}{{\partial \nu_j^{{r + 1}}}}P_{{k,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_{{i - 1}}},\;{\nu_j},\;{\theta_{{i + 1}}},\ldots,\,{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_{{j - 1}}},\;{\nu_{{j + 1}}},\ldots,\;{\nu_m};\;z;\;w),\;h = 1,\;2} \hfill \\ \end{array}\kern0.5em \hfill \\ \kern2.5em \hfill \\\end{array} $$
(A17)
By starting with \( r=0 \) in (A17) and successive substitution of the equation (A17), with \( r=1,2,\ldots,k \) we get
$$ \begin{array}{*{20}c} {P_{{k,\;h}}^{*}({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;z;\;w) = P_{{k,\,h}}^{*(0) }({\theta_1},\;{\theta_2},\ldots,\;{\theta_n};\;{\nu_1},\;{\nu_2},\ldots,\;{\nu_m};\;z;\;w) = } \hfill \\ {\sum\limits_{{l = 1}}^k {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{m\;!(1.2\ldots (k+1-l))}}} \left[ \begin{array}{*{20}c} \sum\limits_{i=1}^n {\frac{{{\lambda_1}z}}{n}{S^{*(k+1-l) }}\left( {{\theta_i}} \right)Q_{l,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_m};z; w\;)} \hfill \\ -\sum\limits_{j=1}^m {\frac{{{\lambda_2}w}}{m}{S^{*(k+1-l) }}\left( {{\nu_j}} \right)Q_{l,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z; w\;)} \hfill \\\end{array} \right]} \hfill \\ {+\sum\limits_{l=1}^k {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{n\;!(1.2\ldots (k+1-l))}}} \sum\limits_{i=1}^n {\left( {1-\frac{{{\lambda_1}z}}{n}{S^{*}}\left( {{\theta_i}} \right)-\frac{(n+1) }{z}} \right)\frac{{{\partial^{k+1-l }}}}{{\partial \theta_i^{k+1-l }}}Q_{l,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_m};z; w\;)} } \hfill \\ {+\sum\limits_{l=1}^k {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{m\;!(1.2\ldots (k+1-l))}}} \sum\limits_{i=1}^n {\left( {1-\frac{{{\lambda_2}w}}{m}{S^{*}}\left( {{\nu_j}} \right)-\frac{(m+1) }{w}} \right)\frac{{{\partial^{k+1-l }}}}{{\partial \nu_j^{k+1-l }}}Q_{l,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z; w\;)} } \hfill \\ { - \sum\limits_{l=1}^{k+1 } {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{m\;!(1.2\ldots (k+1-l))}}} \left[ \begin{array}{*{20}c} \sum\limits_{i=1}^n {\left( {\frac{{l\ !\alpha }}{{\mathrm{n}\;!\mathrm{z}}}} \right)Q_{l,h}^{*(k+1-l) }({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_m};z; w)} \hfill \\ +\sum\limits_{j=1}^m {\left( {\frac{{l\ !\alpha }}{{m!\ \mathrm{w}}}} \right)Q_{l,h}^{*(k+1-l) }({\theta_1},{\theta_2},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z; w\;)} \hfill \\\end{array} \right]} \hfill \\ {-\sum\limits_{l=1}^k {\tfrac{{{(-1)^{k+1-l }}{\beta^{k-1 }}}}{{m\;!(1.2\ldots (k+1-l))}}} \sum\limits_{{\mathrm{i}=1}}^{\mathrm{n}} {\sum\limits_{{\mathrm{j}=1}}^{{\mathrm{m}+\mathrm{i}}} {\frac{{{\lambda_2}}}{\mathrm{m}!}} } P_{l,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{i-1 }},{\nu_j},{\theta_{i+1 }},\ldots,{\theta_n}; {\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m}; z; w\;)\kern0.75em ,\ h=1,2} \hfill \\ \end{array} $$
(A18)
By substituting (A18) in (A15), we have the following recursive form about the boundary function
$$ \begin{array}{*{20}c} \begin{array}{*{20}c} {Q_k^{*}({\theta_1},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+Q_k^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z;w)} \hfill \\ {\left\{ \begin{array}{*{20}c} =\frac{{\frac{{\alpha 2\;!}}{{z\;}}Q_{2,h}^{*}(0;z\;)-{\lambda_1}{P_{1,h }}(z)}}{{1-{\lambda_1}zS*\left( {{\theta_i}} \right)-\frac{2}{z}}}\kern0.75em ,\ k=1, h=1\ (\mathrm{low}\ \mathrm{priority}\ \mathrm{calls}) \hfill \\ =\frac{{\frac{{\alpha 2\;!}}{{w\;}}Q_{2,h}^{*}(0;w\;)-{\lambda_2}{P_{1,h }}(w)}}{{1-{\lambda_2}zS*\left( {{\nu_j}} \right)-\frac{2}{w}}}\kern0.75em ,\ k=1,h=2\kern1em (\mathrm{high}\ \mathrm{priority}\ \mathrm{calls}) \hfill \\ =\frac{{\left( \begin{array}{*{20}c} \left( \begin{array}{*{20}c} \frac{{\alpha (\mathrm{k}+1)!}}{\mathrm{n}!\mathrm{z}}Q_{k+1,h}^{*}({\theta_1},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w) \hfill \\ +\frac{{\alpha (\mathrm{k}+1)!}}{\mathrm{m}!\mathrm{w}}Q_{k+1,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z;w) \hfill \\\end{array} \right) \hfill \\ -\frac{{\beta (\mathrm{k}-1)!}}{\mathrm{m}!\mathrm{n}!}P_{k-1,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+ \hfill \\ \sum\limits_{i=1}^n {\sum\limits_{j=1}^{m+i } {\frac{{{\lambda_2}}}{{m\;!}}S*\left( {{\nu_j}} \right)Q_{k,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{j-1 }},{\nu_j},{\theta_{j+1 }},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z; w)} } \hfill \\\end{array} \right)}}{{\left( {1-\frac{{{\lambda_1}z}}{n}{S^{*}}\left( {{\theta_i}} \right)-\frac{n+1 }{z}} \right)+\left( {1-\frac{{{\lambda_2}w}}{m}{S^{*}}\left( {{\nu_j}} \right)-\frac{m+1 }{w}} \right)}},\kern3.75em \hfill \\ \kern14.25em i = 1,2,\ldots,n,\ j = 1,2,\ldots,m,\ m+n\leq\ C,h=1,2 \hfill \\ =\frac{{\left( \begin{array}{*{20}c} \frac{{\beta (\mathrm{C}-1)!}}{\mathrm{m}!\mathrm{n}!}P_{C-1,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+ \hfill \\ \sum\limits_{i=1}^n {\sum\limits_{j=1}^{m+i } {\frac{{{\lambda_2}}}{{m\;!}}S*\left( {{\nu_j}} \right)Q_{k,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{j-1 }},{\nu_j},{\theta_{j+1 }},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z; w)} } \hfill \\\end{array} \right)}}{{\left( {1-\frac{{{\lambda_1}z}}{n}{S^{*}}\left( {{\theta_i}} \right)-\frac{n+1 }{z}} \right)+\left( {1-\frac{{{\lambda_2}w}}{m}{S^{*}}\left( {{\nu_j}} \right)-\frac{m+1 }{w}} \right)}},\kern0.75em \hfill \\ \kern21em m+n = C,h=1,2 \hfill \\\end{array} \right.} \hfill \\ {Q_{2,h}^{*}(0;z)=\frac{\mathrm{z}}{{\alpha 2\;!\;}}\left( {Q_{1,h}^{*}(0; z\;)\left( {1-{\lambda_1}zS*\left( {{\theta_1}} \right)-\frac{2}{z}} \right) - {\lambda_1}{P_{1,h }}(z)} \right),\ k=1,h=1\ (\mathrm{low}\ \mathrm{priority}\ \mathrm{calls})} \hfill \\ {Q_{2,h}^{*}(0;w)=\frac{\mathrm{w}}{{\alpha 2\;!\;}}\left( {Q_{1,h}^{*}(0; w\;)\left( {1-{\lambda_2}wS*\left( {{\nu_1}} \right)-\frac{2}{w}} \right) - {\lambda_2}{P_{1,h }}(w)} \right),\ k=1,h=2\ (\mathrm{high}\ \mathrm{priority}\ \mathrm{calls})} \hfill \\ \begin{array}{*{20}c} \left( {\frac{m! }{z}Q_{k+1,h}^{*}({\theta_1},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,.{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+\frac{n! }{w}Q_{k+1,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z;w)} \right) \hfill \\ =\frac{m!n! }{{\alpha (k+1)}}\left( {Q_{k,h}^{*}({\theta_1},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+Q_{k,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z;w)} \right) \hfill \\ \kern2em \left[ {\left( {1-\frac{{{\lambda_1}z}}{n}{S^{*}}\left( {{\theta_i}} \right)-\frac{n+1 }{z}} \right)+\left( {1-\frac{{{\lambda_2}w}}{m}{S^{*}}\left( {{\nu_j}} \right)-\frac{m+1 }{w}} \right)} \right]+\tfrac{{\beta (\mathrm{k}-1)!}}{{\alpha (k+1)!}}P_{k-1,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+ \hfill \\ \kern2em \sum\limits_{i=1}^n {\sum\limits_{j=1}^{m+i } {\frac{{n\,!{\lambda_2}}}{{\alpha (k+1)\;!}}S*\left( {{\nu_j}} \right)Q_{k,t}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{j-1 }},{\nu_j},{\theta_{j+1 }},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z; w)} } \hfill \\ \kern25.5em i = 1,2,\ldots, n,\ j = 1,2,\ldots, m,\ m+n\leq\ C,h=1,2 \hfill \\\end{array} \hfill \\ \begin{array}{*{20}c} Q_{k,h}^{*}({\theta_1},\ldots,{\theta_{i-1 }},{\theta_{i+1 }},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+Q_{k,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m};z;w) \hfill \\ \kern1.5em =\frac{{\left( \begin{array}{*{20}c} \frac{{\beta (\mathrm{C}-1)!}}{\mathrm{m}!\mathrm{n}!}P_{C-1,h}^{*}({\theta_1},\ldots,{\theta_n},{\nu_1},\ldots,{\nu_m};z;w)+ \hfill \\ \sum\limits_{i=1}^n {\sum\limits_{j=1}^{m+i } {\frac{{{\lambda_2}}}{{m\;!}}S*\left( {{\nu_j}} \right)P_{C,h}^{*}({\theta_1},{\theta_2},\ldots,{\theta_{j-1 }},{\nu_j},{\theta_{j+1 }},\ldots,{\theta_n};{\nu_1},{\nu_2},\ldots,{\nu_{j-1 }},{\nu_{j+1 }},\ldots,{\nu_m}; z; w)} } \hfill \\\end{array} \right)}}{{\left( {1-\frac{{{\lambda_1}z}}{n}{S^{*}}\left( {{\theta_i}} \right)-\frac{n+1 }{z}} \right)+\left( {1-\frac{{{\lambda_2}w}}{m}{S^{*}}\left( {{\nu_j}} \right)-\frac{m+1 }{w}} \right)}},\kern2.75em \hfill \\ \kern32.5em m+n = C,h=1,2 \hfill \\\end{array} \hfill \\ \end{array} \hfill \\ \kern2.25em \hfill \\ \kern0.5em \hfill \\\end{array} $$
(A19)
