1 Introduction
2 Model
3 Analysis of the \(M_\lambda /M_{\mu _1}^{T_1,B_1}+M_{\mu _2}^{T_2,B_2}/N_1+N_2\) queue
3.1 The joint probability function of the number of busy servers of each type and the queue length
To state | Rates conditional on | ||
---|---|---|---|
\(n_1<N_1\) | \(n_1=N_1\), \(n_2<N_2\) | \(n_1=N_1\), \(n_2=N_2\) | |
\((n_1,n_2,n_Q+1)\) | \(\lambda { 1 }_{\{n_Q+1<T_1\}}\) | \(\lambda { 1 }_{\{n_Q+1<T_2\}}\) | \(\lambda \) |
\((n_1+1,n_2,0)\) | \(\lambda { 1 }_{\{n_Q+1=T_1\}}\) | ||
\((n_1,n_2+1,0)\) | \(\lambda { 1 }_{\{n_Q+1=T_2\}}\) | ||
\((n_1-1,n_2,n_Q)\) | \(n_1\mu _1\) | \(N_{1}\mu _{1}{ 1 }_{\{n_Q<T_1\}}\) | \(N_{1}\mu _{1}{ 1 }_{\{n_Q<T_1\}}\) |
\((n_1,n_2,(n_Q-B_1)^+)\) | \(N_{1}\mu _{1}{ 1 }_{\{n_Q\ge T_1\}}\) | \(N_{1}\mu _{1}{ 1 }_{\{n_Q\ge T_1\}}\) | |
\((n_1,n_2-1,n_Q)\) | \(n_2\mu _2\) | \(n_2\mu _2\) | \(N_{2}\mu _{2}{ 1 }_{\{n_Q<T_2\}}\) |
\((n_1,n_2,(n_Q-B_2)^+)\) | \(N_{2}\mu _{2}{ 1 }_{\{n_Q\ge T_2\}}\) |
3.2 Busy period analysis
3.3 Free period analysis
3.4 Waiting time
3.5 The special case \(T_2=1\)
3.6 The general case \(T_2\ge 1\)
3.6.1 Remaining waiting time
3.6.2 Waiting time
From | To | Transition rate |
---|---|---|
\((1,m,\ell )\quad \)
|
\((1,m,\ell +1)\)
| \({ 1 }_{\{m+\ell <T_1-1\}}\lambda \), |
0 | \({ 1 }_{\{m+\ell \ge T_1-1\}}\lambda \), | |
\((2,m,\ell )\)
|
\((2,m,\ell +1)\)
| \({ 1 }_{\{m+\ell <T_2-1\}}\lambda \), |
\((1,m,\ell )\)
| \({ 1 }_{\{m+\ell <T_1\}}N_{1}\mu _{1}\), | |
\((2,m-B_1,\ell )\)
| \({ 1 }_{\{m>B_1\}}N_{1}\mu _{1}\), | |
0 | \({ 1 }_{\{m+\ell =T_2-1\}}\lambda +{ 1 }_{\{m+\ell \ge T_1\}}{ 1 }_{\{m\le B_1\}}N_{1}\mu _{1}\), | |
\((0,\ell )\)
|
\((0,\ell +1)\)
| \({ 1 }_{\{\ell <T_2-1\}}\lambda \), |
\((1,m,\ell )\)
| \(\alpha ^{m-1}(1-\alpha )N_{1}\mu _{1},\text { for } m=1,\ldots ,T_1-1-\ell \), | |
\((2,m,\ell )\)
| \(\alpha ^{m-1}(1-\alpha )N_{2}\mu _{2},\text { for } m=1,\ldots ,T_2-1-\ell \), | |
0 |
\(\left( \alpha ^{[T_1-\ell -1]^+}-\alpha ^{B_1}\right) N_{1}\mu _{1}+\left( \alpha ^{T_2-\ell -1}-\alpha ^{B_2}\right) N_{2}\mu _{2},\)
|
3.7 Other useful performance measures
4 Analysis of queue with multi-type of servers
From | To | Transition rate |
---|---|---|
\((\sigma ,m,\ell )\) | \((\sigma ,m,\ell +1)\) | \({ 1 }_{\{m+\ell <T_\sigma -1\}}\lambda ,\) |
\((\sigma _1,m,\ell )\) | \({ 1 }_{\{m+\ell <T_{\sigma _1}\}}N_{\sigma _1}\mu _{\sigma _1},\) | |
\((\sigma _1,m-B_{\sigma _1},\ell )\) | \({ 1 }_{\{m>B_{\sigma _1}\}}N_{\sigma _1}\mu _{\sigma _1}\), | |
0 | \(\displaystyle \sum _{\tau =1}^{\sigma -1}{ 1 }_{\{m+\ell \ge T_{\tau }\}}{ 1 }_{\{m\le B_{\tau }\}}N_{\tau }\mu _{\tau }+{ 1 }_{\{m+\ell =T_\sigma -1\}}\lambda \), | |
\((0,\ell )\) | \((0,\ell +1)\) | \({{ 1 }}_{\{\ell <T_\sigma -1\}}\lambda \), |
\((\sigma ,m,\ell )\) | \(\alpha ^{m-1}(1-\alpha )N_{1}\mu _{1},\text { for } m=1,\ldots ,T_\sigma -1-\ell \), | |
0 | \(\displaystyle \sum _{\sigma =1}^S\left( \alpha ^{[T_\sigma -\ell -1]^+}-\alpha ^{B_\sigma }\right) N_{\sigma }\mu _{\sigma },\) |