1. Basic Reaction
Chain Initiation
\[\begin{align} I\xrightarrow{k_d}&2fI^*\\ I^*+M\xrightarrow{k_i}&P_1 \end{align}\]In short:
\[I\xrightarrow{k_d}2fP_1\]Chain Propagation
\[P_n+M\xrightarrow{k_p}P_{n+1}\]Chain Termination
By Combination:
\[P_n+P_m\xrightarrow{k_{t,c}}D_{n+m}\]By Disproportionation:
\[P_n+P_m\xrightarrow{k_{t,d}}D_n+D_m\]Chain Transfer
To Monomer:
\[\begin{align} P_n+M\xrightarrow{k^{mon}_{tr}}&D_n+M^*\\ M^*+M\xrightarrow{k^{mon}_{i}}&P_1 \end{align}\]In short:
\[P_n+M\xrightarrow{k^{mon}_{tr}}D_n+P_1\]To Solvent or Transfer Agent:
\[\begin{align} P_n+S\xrightarrow{k^{sol}_{tr}}&D_n+S^*\\ S^*+S\xrightarrow{k^{sol}_{i}}&P_1 \end{align}\]In short:
\[P_n+S\xrightarrow{k^{sol}_{tr}}D_n+P_1\]This post did not consider the transfer between radicals, the branching in polymerization, and many other side reactions.
2. Radical
2.1 Moment Definition
Define 0th, 1st, and 2nd moment:
\[\begin{align} \lambda_0&=\sum^\infty_{n=1}{P_n}=[P_{tot}]\\ \lambda_1&=\sum^\infty_{n=1}{nP_n}\\ \lambda_2&=\sum^\infty_{n=1}{n^2P_n} \end{align}\]Where Pn is the concentration of radicals with chain length n in certain moment.
2.2 Rate of Change
\[\begin{align} R(P_1)&=\overbrace{2fk_d[I]}^{init}-\overbrace{k_p[M]P_1}^{form\ P_2}-\overbrace{k_t[P_{tot}]P_1}^{term}-\overbrace{k^{mon}_{tr}[M]P_1}^{trans\ to\ mon}-\overbrace{k^{sol}_{tr}[S]P_1}^{trans\ to\ sol}+\overbrace{k^{mon}_{tr}[M][P_{tot}]}^{trans\ from\ mon}+\overbrace{k^{sol}_{tr}[S][P_{tot}]}^{trans\ from\ sol}\\ R(P_2)&=\overbrace{k_p[M]P_1}^{poly\ from\ P_1}-\overbrace{k_p[M]P_2}^{form\ P_3}-\overbrace{k_t[P_{tot}]P_2}^{term}-\overbrace{k^{mon}_{tr}[M]P_2}^{trans\ to\ mon}-\overbrace{k^{sol}_{tr}[S]P_2}^{trans\ to\ sol}\\ &=k_p[M]P_1-k_p[M]P_2-(k_t[P_{tot}]+k^{mon}_{tr}[M]+k^{sol}_{tr}[S])P_2\\ &\vdots\\ R(P_n)_{n>1}&=k_p[M]P_{n-1}-k_p[M]P_n-k_t[P_{tot}]P_n-k^{mon}_{tr}[M]P_n-k^{sol}_{tr}[S]P_n\\ &\vdots \end{align}\]2.3 Moment Equation
\[\begin{align} R(\lambda_0)&=\sum^\infty_{n=1}{R(P_n)}\\ R(\lambda_1)&=\sum^\infty_{n=1}{R(nP_n)}=\sum^\infty_{n=1}{nR(P_n)}\\ R(\lambda_2)&=\sum^\infty_{n=1}{R(n^2P_n)}=\sum^\infty_{n=1}{n^2R(P_n)}\\ \end{align}\]2.3.1 Zeroth Moment $R(\lambda_0)$
\[\begin{align} R(\lambda_0)=&R(P_1)+R(P_2)+\dots+R(P_{n-1})+R(P_n)+\dots\\ =&2fk_d[I]-k_p[M]P_1-k_t[P_{tot}]P_1-k^{mon}_{tr}[M]P_1-k^{sol}_{tr}[S]P_1+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\\ &+k_p[M]P_1-k_p[M]P_2-k_t[P_{tot}]P_2-k^{mon}_{tr}[M]P_2-k^{sol}_{tr}[S]P_2\\ &\vdots\\ &+k_p[M]P_{n-2}-k_p[M]P_{n-1}-k_t[P_{tot}]P_{n-1}-k^{mon}_{tr}[M]P_{n-1}-k^{sol}_{tr}[S]P_{n-1}\\ &+k_p[M]P_{n-1}-k_p[M]P_n-k_t[P_{tot}]P_n-k^{mon}_{tr}[M]P_n-k^{sol}_{tr}[S]P_n\\ &\vdots\\ =&2fk_d[I]+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\\ &+k_p[M](-P_1+P_1-P_2+P_2-\dots -P_{n-1}+P_{n-1}-P_n+P_n+\dots)\\ &-k_t[P_{tot}](P_1+P_2+\dots+P_{n-1}+P_n+\dots)\\ &-k^{mon}_{tr}[M](P_1+P_2+\dots+P_{n-1}+P_n+\dots)\\ &-k^{sol}_{tr}[S](P_1+P_2+\dots+P_{n-1}+P_n+\dots)\\ =&2fk_d[I]+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]-k_t[P_{tot}]^2-k^{mon}_{tr}[M][P_{tot}]-k^{sol}_{tr}[S][P_{tot}]\\ =&2fk_d[I]-k_t[P_{tot}]^2\\ R(\lambda_0)=&2fk_d[I]-k_t\lambda^2_0 \end{align}\]2.3.2 First Moment $R(\lambda_1)$
\[\begin{align} R(\lambda_1)=&1R(P_1)+2R(P_2)+\dots+(n-1)R(P_{n-1})+nR(P_n)+\dots\\ =&2fk_d[I]-k_p[M]P_1-k_t[P_{tot}]P_1-k^{mon}_{tr}[M]P_1-k^{sol}_{tr}[S]P_1+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\\ &+2k_p[M]P_1-2k_p[M]P_2-2k_t[P_{tot}]P_2-2k^{mon}_{tr}[M]P_2-2k^{sol}_{tr}[S]P_2\\ &\vdots\\ &+(n-1)k_p[M]P_{n-2}-(n-1)k_p[M]P_{n-1}-(n-1)k_t[P_{tot}]P_{n-1}-(n-1)k^{mon}_{tr}[M]P_{n-1}-(n-1)k^{sol}_{tr}[S]P_{n-1}\\ &+nk_p[M]P_{n-1}-nk_p[M]P_n-nk_t[P_{tot}]P_n-nk^{mon}_{tr}[M]P_n-nk^{sol}_{tr}[S]P_n\\ &\vdots\\ =&2fk_d[I]+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\\ &+k_p[M](-P_1+2P_1-2P_2+3P_2-\dots -(n-2)P_{n-1}+(n-1)P_{n-1}-(n-1)P_n+nP_n+\dots)\\ &-k_t[P_{tot}]\overbrace{(P_1+2P_2+\dots+(n-1)P_{n-1}+nP_n+\dots)}^{\lambda_1}\\ &-k^{mon}_{tr}[M](P_1+2P_2+\dots+(n-1)P_{n-1}+nP_n+\dots)\\ &-k^{sol}_{tr}[S](P_1+2P_2+\dots+(n-1)P_{n-1}+nP_n+\dots)\\ =&2fk_d[I]+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\\ &+k_p[M](P_1+P2+\dots+P_{n-1}+P_n+\dots)\\ &-k_t\lambda_0\lambda_1-k^{mon}_{tr}[M]\lambda_1-k^{sol}_{tr}[S]\lambda_1\\ =&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0+k_p[M]\lambda_0-k_t\lambda_0\lambda_1-k^{mon}_{tr}[M]\lambda_1-k^{sol}_{tr}[S]\lambda_1\\ R(\lambda_1)=&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0+k_p[M]\lambda_0-(k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S])\lambda_1 \end{align}\]2.3.3 Second Moment $R(\lambda_2)$
\[\begin{align} R(\lambda_2)=&1^2R(P_1)+2^2R(P_2)+\dots+(n-1)^2R(P_{n-1})+n^2R(P_n)+\dots\\ =&2fk_d[I]-k_p[M]P_1-k_t[P_{tot}]P_1-k^{mon}_{tr}[M]P_1-k^{sol}_{tr}[S]P_1+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\\ &+2^2k_p[M]P_1-2^2k_p[M]P_2-2^2k_t[P_{tot}]P_2-2^2k^{mon}_{tr}[M]P_2-2^2k^{sol}_{tr}[S]P_2\\ &\vdots\\ &+(n-1)^2k_p[M]P_{n-2}-(n-1)^2k_p[M]P_{n-1}-(n-1)^2k_t[P_{tot}]P_{n-1}-(n-1)^2k^{mon}_{tr}[M]P_{n-1}-(n-1)^2k^{sol}_{tr}[S]P_{n-1}\\ &+n^2k_p[M]P_{n-1}-n^2k_p[M]P_n-n^2k_t[P_{tot}]P_n-n^2k^{mon}_{tr}[M]P_n-n^2k^{sol}_{tr}[S]P_n\\ &\vdots\\ =&2fk_d[I]+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\\ &-k_t[P_{tot}]\overbrace{(P_1+2^2P_2+\dots+(n-1)^2P_{n-1}+n^2P_n+\dots)}^{\lambda_2}\\ &-k^{mon}_{tr}[M](P_1+2^2P_2+\dots+(n-1)^2P_{n-1}+n^2P_n+\dots)\\ &-k^{sol}_{tr}[S](P_1+2^2P_2+\dots+(n-1)^2P_{n-1}+n^2P_n+\dots)\\ &+k_p[M](-1^2P_1+2^2P_1-2^2P_2+\dots+(n-1)^2P_{n-2}-(n-1)^2P_{n-1}+n^2P_{n-1}-n^2P_n+\dots)\\ =&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0-k_t\lambda_0\lambda_2-k^{mon}_{tr}[M]\lambda_2-k^{sol}_{tr}[S]\lambda_2\\ &+k_p[M](\sum^\infty_{n=1}{(n+1)^2P_n}-\sum^\infty_{n=1}{n^2P_n})\\ =&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0-(k_t\lambda_0-k^{mon}_{tr}[M]-k^{sol}_{tr}[S])\lambda_2\\ &+k_p[M](\sum^\infty_{n=1}{(n+1)^2P_n}-\sum^\infty_{n=1}{n^2P_n})\\ =&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0-(k_t\lambda_0-k^{mon}_{tr}[M]-k^{sol}_{tr}[S])\lambda_2\\ &+k_p[M](\sum^\infty_{n=1}{n^2P_n}+2\sum^\infty_{n=1}{nP_n}+\sum^\infty_{n=1}{P_n}-\sum^\infty_{n=1}{n^2P_n})\\ R(\lambda_2)=&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0+k_p[M](2\lambda_1+\lambda_0)-(k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S])\lambda_2 \end{align}\]2.4 Radical Distribution
For a constant volume batch reactor:
We have
\[\begin{align} {d\lambda_0\over dt}=&R(\lambda_0)\\ {d\lambda_1\over dt}=&R(\lambda_1)\\ {d\lambda_2\over dt}=&R(\lambda_2) \end{align}\]If we apply QSSA to get the differentiation equation:
Since QSSA, $R(\lambda_0)=R(\lambda_1)=R(\lambda_2)=0$,
\[\begin{align} R(\lambda_0)=2fk_d[I]-k_t\lambda^2_0=&0\\ k_t\lambda^2_0=&2fk_d[I]\\ [P_{tot}]=\lambda_0=&\sqrt{2fk_d[I]\over k_t} \end{align}\]As we defined $\tau,\ \beta$:
\[\begin{align} \tau =&{k_{t,d}\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S]\over k_p[M]}\\ \beta =&{k_{t,c}\lambda_0\over k_p[M]} \end{align}\]And
\[\begin{align} R(\lambda_1)=&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0+k_p[M]\lambda_0-(k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S])\lambda_1=0\\ \lambda_1=&{2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0+k_p[M]\lambda_0\over k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S]}\\ \lambda_1=&{(k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S])\lambda_0+k_p[M]\lambda_0\over k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S]}\\ \lambda_1=&({\tau +\beta +1\over \tau +\beta})\lambda_0 \end{align}\]AS $DP^{rad}_{n}={\lambda_1\over \lambda_0}$
\[DP^{rad}_{n}={\tau +\beta +1\over \tau +\beta}\]And apply LCH, where is The consumption of monomer by chain-initiation or transfer events is negligible compared to that by propagation:
\[\begin{align} \tau +\beta +1\approx &1\\ DP^{rad}_{n}=&{1\over \tau +\beta} \end{align}\]Since
\[\begin{align} R(\lambda_2)=&2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0+k_p[M](2\lambda_1+\lambda_0)-(k_t\lambda_0-k^{mon}_{tr}[M]-k^{sol}_{tr}[S])\lambda_2=0\\ \lambda_2=&{2fk_d[I]+k^{mon}_{tr}[M]\lambda_0+k^{sol}_{tr}[S]\lambda_0+k_p[M](2\lambda_1+\lambda_0)\over k_t\lambda_0-k^{mon}_{tr}[M]-k^{sol}_{tr}[S]}\\ \lambda_2=&{(k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S])\lambda_0+k_p[M](2\lambda_1+\lambda_0)\over k_t\lambda_0-k^{mon}_{tr}[M]-k^{sol}_{tr}[S]}\\ \lambda_2=&{2\lambda_1+(1+\tau +\beta)\lambda_0\over \tau +\beta} \end{align}\]As
\[\lambda_1=({\tau +\beta +1\over \tau +\beta})\lambda_0\]So
\[\lambda_2={2+\tau +\beta \over \tau +\beta}\lambda_1\]AS $DP^{rad}_{w}={\lambda_2\over \lambda_1}$
\[DP^{rad}_{w}={2+\tau +\beta \over \tau +\beta}\]And apply LCH,
\[DP^{rad}_{w}={2\over \tau +\beta}\]3. Flory–Schulz Distribution
The probability of propagation is defined as:
\[P={1\over 1+\tau +\beta}\]Therefore
\[\begin{align} DP^{rad}_{n}=&{1\over 1-P}\\ DP^{rad}_{w}=&{1+P\over 1-P} \end{align}\]If P is approaching 1,
\[PDI={DP^{rad}_{w}\over DP^{rad}_{n}}={1+P\over 1}=2\] \[R(P_n)_{n>1}=k_p[M]P_{n-1}-k_p[M]P_n-k_t[P_{tot}]P_n-k^{mon}_{tr}[M]P_n-k^{sol}_{tr}[S]P_n\]Apply QSSA,
\[\begin{align} R(P_n)_{n>1}=&k_p[M]P_{n-1}-k_p[M]P_n-k_t[P_{tot}]P_n-k^{mon}_{tr}[M]P_n-k^{sol}_{tr}[S]P_n=0\\ P_n=&{k_p[M]P_{n-1}\over k_p[M]-k_t\lambda_0-k^{mon}_{tr}[M]-k^{sol}_{tr}[S]}\\ P_n=&{1\over 1+\tau +\beta}P_{n-1} \end{align}\]So
\[P_n=({1\over 1+\tau +\beta})^{n-1}P_1 \Leftarrow \begin{cases} P_{n-1}=&{1\over 1+\tau +\beta}P_{n-2}\\ P_{n-2}=&{1\over 1+\tau +\beta}P_{n-3}\\ &\vdots \end{cases}\]And
\[\begin{align} R(P_1)&=2fk_d[I]-k_p[M]P_1-k_t[P_{tot}]P_1-k^{mon}_{tr}[M]P_1-k^{sol}_{tr}[S]P_1+k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]=0\\ P_1&={(k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S])\lambda_0\over k_p[M]+k_t\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S]}\\ P_1&={\tau +\beta \over 1+\tau +\beta}\lambda_0 \end{align}\]Hence,
\[\begin{align} P_n=&({1\over 1+\tau +\beta})^{n-1}({\tau +\beta \over 1+\tau +\beta})\lambda_0\\ =&P^{n-1}(1-P)\lambda_0 \end{align}\]And the fraction of polymer chain has repeat unit of n
\[f(n)={P_n\over \lambda_0}=P^{n-1}(1-P)\]
4. Product
4.1 Moment Definition
Define 0th, 1st, and 2nd moment:
\[\begin{align} \mu_0&=\sum^\infty_{n=1}{D_n}\\ \mu_1&=\sum^\infty_{n=1}{nD_n}\\ \mu_2&=\sum^\infty_{n=1}{n^2D_n} \end{align}\]Where Dn is the concentration of dead chain with chain length n in certain moment.
4.2 Rate of Change
\[\begin{align} R(D_1)=&k_{t,d}\lambda_0P_1+k^{sol}_{tr}[S]P_1+k^{mon}_{tr}[M]P_1\\ R(D_n)=&\overbrace{k_{t,d}\lambda_0P_n}^{term\ by\ dispropotion}+\overbrace{k^{sol}_{tr}[S]P_n}^{sol\ trans}+\overbrace{k^{mon}_{tr}[M]P_n}^{mon\ trans}+\overbrace{0.5k_{t,c}\sum^{n-1}_{m=1}{P_mP_{n-m}}}^{term\ by\ combination} \end{align}\]4.3 Moment Equation
\[\begin{align} R(\mu_0)&=\sum^\infty_{n=1}{R(D_n)}\\ R(\mu_1)&=\sum^\infty_{n=1}{R(nD_n)}=\sum^\infty_{n=1}{nR(D_n)}\\ R(\mu_2)&=\sum^\infty_{n=1}{R(n^2D_n)}=\sum^\infty_{n=1}{n^2R(D_n)} \end{align}\]4.3.1 Zeroth Moment $R(\mu_0)$
\[\begin{align} R(\mu_0)=&R(D_1)+R(D_2)+R(D_3)+\dots+R(D_{n-1})+R(D_n)+\dots\\ =&k_{t,d}\lambda_0P_1+k^{sol}_{tr}[S]P_1+k^{mon}_{tr}[M]P_1\\ &+k_{t,d}\lambda_0P_2+k^{sol}_{tr}[S]P_2+k^{mon}_{tr}[M]P_2+0.5k_{t,c}\sum^{2}_{m=1}{P_mP_{2-m}}\\ &+k_{t,d}\lambda_0P_3+k^{sol}_{tr}[S]P_3+k^{mon}_{tr}[M]P_3+0.5k_{t,c}\sum^{3}_{m=1}{P_mP_{3-m}}\\ &\vdots\\ &+k_{t,d}\lambda_0P_{n-1}+k^{sol}_{tr}[S]P_{n-1}+k^{mon}_{tr}[M]P_{n-1}+0.5k_{t,c}\sum^{n-1}_{m=1}{P_mP_{n-1-m}}\\ &+k_{t,d}\lambda_0P_n+k^{sol}_{tr}[S]P_n+k^{mon}_{tr}[M]P_n+0.5k_{t,c}\sum^{n}_{m=1}{P_mP_{n-m}}\\ &\vdots\\ =&k_{t,d}\lambda_0^2+k^{sol}_{tr}[S]\lambda_0+k^{mon}_{tr}[M]\lambda_0\\ &+0.5k_{t,c}[P_1P_1+\\ &\qquad\qquad P_1P_2+P_2P_1+\\ &\qquad\qquad\vdots\\ &\qquad\qquad P_1P_{n-2}+P_2P_{n-3}+\dots+P_{n-3}P_2+P_{n-2}P_1\\ &\qquad\qquad P_1P_{n-1}+P_2P_{n-2}+P_3P_{n-3}+\dots+P_{n-3}P_3+P_{n-2}P_2+P_{n-1}P_1\\ &\qquad\qquad \vdots\qquad]\\ =&k_{t,d}\lambda_0^2+k^{sol}_{tr}[S]\lambda_0+k^{mon}_{tr}[M]\lambda_0\\ &+0.5k_{t,c}(P_1\lambda_0+P_2\lambda_0+\dots P_{n-1}\lambda_0+P_n\lambda_0+\dots)\\ R(\mu_0)=&k_{t,d}\lambda_0^2+k^{sol}_{tr}[S]\lambda_0+k^{mon}_{tr}[M]\lambda_0+0.5k_{t,c}\lambda_0^2 \end{align}\]4.3.2 First Moment $R(\mu_1)$
\[\begin{align} R(\mu_1)=&1R(D_1)+2R(D_2)+3R(D_3)+\dots+(n-1)R(D_{n-1})+nR(D_n)+\dots\\ =&k_{t,d}\lambda_0P_1+k^{sol}_{tr}[S]P_1+k^{mon}_{tr}[M]P_1\\ &+2k_{t,d}\lambda_0P_2+2k^{sol}_{tr}[S]P_2+2k^{mon}_{tr}[M]P_2+2*0.5k_{t,c}\sum^{2}_{m=1}{P_mP_{2-m}}\\ &+3k_{t,d}\lambda_0P_3+3k^{sol}_{tr}[S]P_3+3k^{mon}_{tr}[M]P_3+3*0.5k_{t,c}\sum^{3}_{m=1}{P_mP_{3-m}}\\ &\vdots\\ &+(n-1)k_{t,d}\lambda_0P_{n-1}+(n-1)k^{sol}_{tr}[S]P_{n-1}+(n-1)k^{mon}_{tr}[M]P_{n-1}+(n-1)0.5k_{t,c}\sum^{n-1}_{m=1}{P_mP_{n-1-m}}\\ &+nk_{t,d}\lambda_0P_n+nk^{sol}_{tr}[S]P_n+nk^{mon}_{tr}[M]P_n+n0.5k_{t,c}\sum^{n}_{m=1}{P_mP_{n-m}}\\ &\vdots\\ =&k_{t,d}\lambda_0\lambda_1+k^{sol}_{tr}[S]\lambda_1+k^{mon}_{tr}[M]\lambda_1\\ &+0.5k_{t,c}[2P_1P_1+\\ &\qquad\qquad 3P_1P_2+3P_2P_1+\\ &\qquad\qquad\vdots\\ &\qquad\qquad (n-1)P_1P_{n-2}+(n-1)P_2P_{n-3}+\dots+(n-1)P_{n-3}P_2+(n-1)P_{n-2}P_1\\ &\qquad\qquad nP_1P_{n-1}+nP_2P_{n-2}+nP_3P_{n-3}+\dots+nP_{n-3}P_3+nP_{n-2}P_2+nP_{n-1}P_1\\ &\qquad\qquad \vdots\qquad]\\ =&k_{t,d}\lambda_0\lambda_1+k^{sol}_{tr}[S]\lambda_1+k^{mon}_{tr}[M]\lambda_1\\ &+0.5k_{t,c}[P_1(2P_1+3P_2+\dots+nP_{n-1}+\dots)+\\ &\qquad\qquad P_2(3P_1+4P_2+\dots++nP_{n-2}+\dots)+\\ &\qquad\qquad\vdots\\ &\qquad\qquad P_{n-1}(nP_1+(n+1)P_2+\dots+(2n-1)P_n)+\\ &\qquad\qquad \vdots\qquad]\\ =&k_{t,d}\lambda_0\lambda_1+k^{sol}_{tr}[S]\lambda_1+k^{mon}_{tr}[M]\lambda_1\\ &+0.5k_{t,c}(P_1\sum^{\infty}_{n=1}{(n+1)P_n}+P_2\sum^{\infty}_{n=1}{(n+2)P_n}+\dots+P_m\sum^{\infty}_{n=1}{(n+m)P_n}+\dots)\\ =&k_{t,d}\lambda_0\lambda_1+k^{sol}_{tr}[S]\lambda_1+k^{mon}_{tr}[M]\lambda_1+0.5k_{t,c}(\lambda_0\lambda_1+\lambda_1\lambda_0)\\ R(\mu_1)=&k_{t,d}\lambda_0\lambda_1+k^{sol}_{tr}[S]\lambda_1+k^{mon}_{tr}[M]\lambda_1+k_{t,c}\lambda_0\lambda_1\\ \end{align}\]4.3.3 Second Moment $R(\mu_2)$
\[\begin{align} R(\mu_2)=&1R(D_1)+2^2R(D_2)+3^2R(D_3)+\dots+(n-1)^2R(D_{n-1})+n^2R(D_n)+\dots\\ =&k_{t,d}\lambda_0\lambda_2+k^{sol}_{tr}[S]\lambda_2+k^{mon}_{tr}[M]\lambda_2+0.5k_{t,c}\sum^{\infty}_{n=1}{n^2\sum^{n-m}_{m=1}{P_mP_{n-m}}}\\ R(\mu_2)=&k_{t,d}\lambda_0\lambda_2+k^{sol}_{tr}[S]\lambda_2+k^{mon}_{tr}[M]\lambda_2+0.5k_{t,c}(\lambda_0\lambda_2+\lambda_1^2) \end{align}\]4.4 Product Distribution
For cumulative distribution:
\[\begin{align} {DP}^{cum}_n&={\mu_1\over \mu_0}\\ {DP}^{cum}_w&={\mu_2\over \mu_1} \end{align}\]For instantaneous distribution:
\[\begin{align} {DP}^{inst}_n&={R(\mu_1)\over R(\mu_0)}\\ {DP}^{inst}_w&={R(\mu_2)\over R(\mu_1)} \end{align}\]As we defined $\tau,\ \beta$:
\[\begin{align} \tau =&{k_{t,d}\lambda_0+k^{mon}_{tr}[M]+k^{sol}_{tr}[S]\over k_p[M]}\\ \beta =&{k_{t,c}\lambda_0\over k_p[M]} \end{align}\]Hence,
\[\begin{align} R(\mu_0)=&k_p[M]\lambda_0(\tau+0.5\beta)\\ R(\mu_1)=&k_p[M]\lambda_1(\tau+\beta)\\ R(\mu_2)=&k_p[M]\lambda_2(\tau+\beta)+k_{t,c}\lambda_1^2\\ \end{align}\]For DPninst:
\[\begin{align} {DP}_n^{inst}=&{R(\mu_1)\over R(\mu_0)}\\ =&{\lambda_1 \over \lambda_0}*{\tau+\beta\over \tau+0.5\beta} \end{align}\]Since:
\[\begin{align} {DP}^{rad}_n=&{\lambda_1 \over \lambda_0}\\ =&{\tau +\beta +1\over \tau +\beta} \end{align}\]So,
\[\begin{align} {DP}_n^{inst}=&{\tau +\beta +1\over \tau +\beta}*{\tau+\beta\over \tau+0.5\beta}\\ =&{\tau +\beta +1\over \tau+0.5\beta} \end{align}\]Apply LCH to the equation:
\[{DP}_n^{inst}\approx {1\over \tau+0.5\beta}\]For DPwinst:
\[\begin{align} {DP}_w^{inst}=&{R(\mu_2)\over R(\mu_1)}\\ =&{k_p[M]\lambda_2(\tau+\beta)+k_{t,c}\lambda_1^2\over k_p[M]\lambda_1(\tau+\beta)}\\ =&{\lambda_2\over \lambda_1}+{k_{t,c}\lambda_1^2\over k_p[M]\lambda_1(\tau+\beta)} \end{align}\]Since:
\[\begin{align} {DP}^{rad}_w=&{\lambda_2 \over \lambda_1}\\ =&{2+\tau +\beta \over \tau +\beta}\\ \\ \beta =&{k_{t,c}\lambda_0\over k_p[M]}\\ {k_{t,c}\over k_p[M]}=&{\beta\over \lambda_0} \end{align}\]So,
\[\begin{align} {DP}_w^{inst}=&{2+\tau +\beta \over \tau +\beta}+{\beta\lambda_1\over \lambda_0(\tau+\beta)}\\ =&{2+\tau+\beta\over \tau+\beta}+{\lambda_1\over \lambda_0}{\beta\over \tau+\beta}\\ =&{2+\tau+\beta\over \tau+\beta}+{1+\tau+\beta\over \tau+\beta}*{\beta\over \tau+\beta} \end{align}\]Apply LCH to the equation:
\[\begin{align} {DP}_w^{inst}\approx&{2\over \tau+\beta}+{\beta\over (\tau+\beta)^2}\\ =&{2\tau+3\beta\over (\tau+\beta)^2} \end{align}\]