## Appendix A Steps for Determining Adjustment of Offset  
_Step 1:_ If, at the end of the cycle, the signal at the upstream intersection is in the green phase for the downstream intersection's through direction, then estimate the number of queued through vehicles in the transition section at the time when the green phase ends by  
\[Velt_{l_{d},k,s}^{det}= \begin{cases}Velt_{l_{d},k,s}^{re}(t)-\left(g_{l_{d},k, s}^{e}(t)-t(n)\right)\times n_{l_{d},s}^{lane}/\overline{h_{t}}&,g_{l_{d},k,s}^{b}(t) \leq t(n)\leq g_{l_{d},k,s}^{e}(t)(a)\\ Velt_{l_{d},k,s}^{e}(t)&,g_{l_{d},k,s}^{e}(t)<t(n)<g_{l_{d},k+1,s}^{b}\ \ (b) \end{cases}  \tag{1}\]  
where \(Velt_{l_{d},k,s}^{det}=\) total number of queued vehicles in the downstream intersection's through direction in cycle \(k\), \(Velt_{l_{d},k,s}^{re}(t)=\) remaining number of vehicles in the downstream intersection's through direction at the current moment in cycle \(k\), \(t(n)=\) moment when the signal phase of the upstream intersection's cycle ends, \(n_{l_{d},s}^{lane}=\) number of through lanes at the downstream intersection, and \(\overline{h_{t}}=\) average headway between vehicles. Note that this estimation is based on the remaining number of through vehicles on the transition section and the number of vehicles that can exit from the downstream intersection within the remaining green phase time (Equation (1a)). If, at the end of the cycle, the signal phase at the upstream intersection is in the red phase for the downstream intersection's through direction, then the current remaining number of through vehicles on the transition section is considered the number of queued vehicles (Equation (1b)). _Step 2:_ The propagation time of the starting wave for queued vehicles in each lane of the westbound approach at the downstream intersection is calculated as  
\[\left\{ \begin{array}{l}L_{l_{d},k+1,s}^{q}=Velt_{l_{d},k,s}^{det}/n_{l_{d}, s}^{lane}\times\overline{h}_{d}\\ t_{l_{d},k+1,s}^{tr}=L_{l_{d},k+1,s}^{q}\times 3.6/V_{sw}\end{array} \right. \tag{2}\]  
where \(L_{l_{d},k+1,s}^{q}=\) length of queued vehicles in a single lane of the westbound through direction at the downstream intersection in cycle \(k+1\), \(\overline{Velt}_{L,k+1,s}^{lane}=\) average number of queued vehicles from the previous cycle that need to pass through each lane in the westbound through direction at the downstream intersection in cycle \(k+1\), \(\overline{h}_{d}=\) average headway between vehicles in a saturated traffic state, and \(t_{l_{d},k+1,s}^{tr}=\) time it takes for the starting wave of queued vehicles in the westbound through direction at the downstream intersection to propagate in cycle \(k+1\). _Step 3:_ The offset is adjusted so that when the upstream intersection's through traffic reaches the end of the queue at the downstream intersection, the phase start wave propagates to the last queued vehicle. This allows the arriving through traffic to follow closely behind the queued vehicles. Thus, this adjustment is given by  
\[\left\{ \begin{array}{l}t_{l_{u},k+1,s}^{hc,arr}=\left(L_{g}-L_{l_{d},k+1,s}^{ q}\right)\times 3.6/V_{l_{u},k+1,s}^{des}\\ T_{k+1,os}^{new}=t_{l_{u},k+1,s}^{hc,arr}-t_{l_{d},k+1,s}^{tr}\end{array} \right. \tag{11}\]  
where \(t_{l_{u},k+1,s}^{hc,arr}=\) time it takes for the first vehicle in the upstream intersection's through traffic to reach the tail end of the queued vehicles at the downstream intersection in cycle \(k+1\), \(V_{k+1,s}^{des}=\) expected speed of the merging through traffic at the upstream intersection in cycle \(k+1\), \(T_{k+1,os}^{new}=\) new offset after adjustment in cycle \(k+1\). _Step 4:_ The adjustment of the offset \(\Delta T_{k+1,os}\) is the difference between the offsets after and before adjustment,  
\[\Delta T_{k+1,os}=T_{k+1,os}^{new}-T_{k+1,os}^{old} \tag{12}\]  
where \(T_{k+1,os}^{new}=\) offset adjusted for cycle \(k+1\), \(\Delta T_{k+1,os}=\) offset adjustment amount, and \(T_{k+1,os}^{old}=\) unadjusted offset for cycle \(k+1\).