Fundamentals of Transportation/Operations

Capacitated (finite) queues (M/M/1) (random arrival and random service), with capacity N

Probability of n units in the system $P(n)=\frac{{\rho }^n\left(1-\rho \right)}{1-{\rho }^{N+1}}$ Expected number of units in the system $E(n)=\frac{\left(\rho \right)}{{\left(1-\rho \right)}^{}}\frac{1-\left(N+1\right){\left(\rho \right)}^N+N{\rho }^{N+1}}{1-{\rho }^{N+1}}$

Gaps

$P(x)=\frac{\lambda t^x{e}^{-\lambda t}}{x!}forx=0,1,2,\mathrm{\infty }$

Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array}{l} {\rm{P(x = 0) = }}\lambda {\rm{e}}^{{\rm{ - }}\lambda {\rm{t}}} \\ {\rm{Freq (h \ge t) = (V - 1) e}}^{{\rm{ - }}\lambda {\rm{t}}} \\ {\rm{Freq (h \lt t) = (V - 1) (1 - e}}^{{\rm{ - }}\lambda {\rm{t}}} {\rm{)}} \\ \\ \end{array} }

Level of Service

$PHF=V/(4V_{15})$ where: $V=HourlyVolume$

$SF=V/PHF=4V_{15}$

$MSF=c_j(v/c{)}_i$

$S{F}_i=MS{F}_{i}N{f}_w{f}_{HV}{f}_p$

$S{F}_i=c_j(v/c{)}_{i}N{f}_w{f}_{HV}{f}_p$

$f_{HV}=\frac{1}{1+P_T\left(E_T-1\right)+P_R\left(E_R-1\right)}$

$FFS=FFSi-f_{LW}-f_{LC}-f_M-f_A$

$TLC=L{C}_R+L{C}_L$

$NAPM\le 40:f_A=0.25NAPM$

$NAPM>40;f_A=10$

$v_p^{}=\frac{V}{PHFN{f}_{HV}{f}_p}$

$S{F}_i=2800{\left(\frac{v}{c}\right)}_i{f}_d{f}_w{f}_{HV}$

$K=DHV/AADT$

$f_{HV}=\frac{1}{1+P_T\left(E_T-1\right)+P_R\left(E_R-1\right)+P_B\left(E_B-1\right)}$

Traffic Signals

$t_c=\frac{\rho r}{1-\rho }$

$P_q=\frac{r+t_c}{C}$

$P_s=\frac{\lambda \left(r+t_C\right)}{\lambda \left(r+g\right)}=\frac{r+t_C}{C}=P_q$

$P_s=\frac{\lambda \left(r+t_C\right)}{\lambda \left(r+g\right)}=\frac{\mu t_C}{\lambda C}=\frac{\mu t_C}{\rho C}$

$Q_{\max }=\lambda r$

$D_t=\frac{\lambda r^2}{2\left(1-\rho \right)}$

$d_{avg}=\frac{\lambda r^2}{2\left(1-\rho \right)}\frac{1}{\lambda C}$

$$

$d_{avg}=\frac{r^2}{2C\left(1-\rho \right)}$

$d_{\max }=r$

$c=s\frac{g}{C}$

$d=\left(d_{1}PF\right)+d_2+d_3$

$d_1=\frac{0.5C{\left(1-\frac{g}{C}\right)}^2}{1-\left[\min \left(1,X\right)\frac{g}{C}\right]}$

$d_2=900T\left[\left(X-1\right)+\sqrt{{\left(X-1\right)}^2+\frac{8kIX}{cT}}\right]$

$d_A=\frac{\sum _i{d}_i{v}_i}{\sum _i{v}_i}$

$d_I=\frac{\sum _A{d}_A{v}_A}{\sum _A{v}_A}$

$Y_c=\sum _{i=1}^n{\left(\frac{v}{s}\right)}_{ci}$

$L=\sum _{i=1}^n{\left(t_L\right)}_{ci}$

$C_{\min }=\frac{L*X_c}{{\mathrm{X}}_{\mathrm{c}}\mathrm{-}\sum _{\mathrm{i}=\mathrm{1}}^{\mathrm{n}}\mathrm{Y}\mathrm{i}}$

$C_{opt}=\frac{\left[\left(1.5L\right)+5\right]}{\left(1.0-\sum _{i=1}^n{Y}_i\right)}$

$C_{\min }=\frac{L{X}_c}{\left(X_c-\sum Y_i\right)}$

$X_i=\frac{v_i}{c_i}=\frac{v_i}{s_i*g_i/C}=\frac{v_i/s_i}{g_i/C}$

$\sum g_i=\sum \frac{v_i}{s_i}\frac{C}{X_c}=C-L$

$X_c=\sum \frac{v_i}{s_i}\frac{C}{C-L}$