We analytically derive approximations for the stop list length n in the steady state, by solving the evolution equation for the length of the planned route after r insertions L_{r}.
$$ L_{r+1}=L_{r}+l^{+}(n)  \nu \Delta t, $$
(2)
Where l^{+}(n) is the average added length per request, νΔt is the distance driven in between requests and the discrete time parameter r counts the requests.
The planned route length reaches its equilibrium at
$$ l^{+}(n) = \nu \Delta t =\frac{2\langle{l}\rangle}{x}, $$
(3)
where we used the definition of x from Eq. 1.
In case of a taxi system the added length is independent of n at two times the average shortest path length in the network
$$ l^{+}_{taxi}= 2 \langle{l}\rangle, $$
(4)
as new segments are simply added to the end. If x>1, the system no longer has an equilibrium as the route length keeps growing, if on the other hand x<1, the taxi has time between subsequent rides, in which it stands still, lowering the average velocity.
In a ridesharing system with a sensible dispatcher algorithm, however the length of the added segments depends on the planned route. In our model for example, as the current length of the route n grows, the probability that a new request’s pickup and/or dropoff being already included in the route increases, resulting in a smaller l^{+}(n).
We take the added length to be the average over three possibilities:

a
Both, pickup and dropoff node are already on the route.

b
The pickup node is on the route but the dropoff node is not.

c
The pickup node is not on the route.
In case a) no length is added to the route, in case b) the average added length is 〈l〉 and in case c) the route gets longer by an average of 2〈l〉. This means that
$$ l^{+}(n) \approx P_{b} \langle{l}\rangle+2 P_{c} \langle{l}\rangle, $$
(5)
where P_{b} and P_{c} are the probabilities of b and c respectively.
To evaluate the probabilities, we introduce route volumeV as the number of nodes that can be reached within the stop list without a detour (illustrated in Fig. 7 in Appendix A). The probability for the requested pickup node to be on the route depends on the expectation value 〈V〉 of the volume of the route. The volume depends on the length of the stoplist as well as the topology of the underlying network. As the requested dropoff point has to be on the route after the pickup, there is a second relevant route volume V_{rest}, and its expectation value 〈V_{rest}〉(n), the average volume of the route after the pickup. Assuming the position of the pickup is uniformly randomly located somewhere along the route, the fact that the insertion of the dropoff is always after the pickup leads to (we employ a simplifying assumption that the pickup is equally likely to be inserted at any position in the stoplist)
$$ \langle{V}\rangle_{\textsf{rest}}(n) \approx \sum_{k=1}^{n} \langle{V}\rangle(nk)/n={\sum_{k=0}^{n1} \langle{V}\rangle(k)/n.} $$
(6)
We see in Fig. 2ab that V_{rest} so computed is slightly lower than the actual value. This is because in reality, the pickup location is slightly more likely to be at the second half of the stop list than the first half. We demonstrate this in Fig. 9 in Appendix A.
Furthermore, we note that the function 〈V〉(n) is always monotonously growing with n and asymptotically approaching N. We thus express 〈V〉_{rest}(n) for large n as
$$ \langle{V}\rangle_{\textsf{rest}}(n) = \frac{1}{n} \sum_{k=0}^{n1} \left[N(N\langle{V}\rangle(k))\right] = N  \frac{1}{n}\sum_{k=0}^{n1} \left[N\langle{V}\rangle(k)\right] \approx N\left(1\frac{\alpha}{n}\right), $$
(7)
where \(\alpha =\sum _{k=0}^{n1} \left [1\langle {V}\rangle (k)/N\right ]\), if this limit exists. Note that \({\lim }_{n\to \infty }1\langle {V}\rangle (n)/N=0\), and if in addition we know that 1−〈V〉(n)/N goes to 0 faster than 1/n, then α is guaranteed to exist. In this case, α is a constant, depending only on the volume growth in a particular network, so 〈V〉_{rest} approaches N with n^{−1}. We demonstrate in Fig. 2c that at least for rings and stars, this assumption holds.
Using this, we can express the probabilities for the three insertion types:
$$\begin{array}{*{20}l} P_{a} &= \frac{\langle{V}\rangle(n)}{N}\frac{\langle{V}\rangle_{\textsf{rest}}(n)}{N} \\ P_{b} &= \frac{\langle{V}\rangle(n)}{N} \left(1 \frac{\langle{V}\rangle_{\textsf{rest}}(n)}{N}\right) \\ P_{c} &= 1 \frac{\langle{V}\rangle(n)}{N}. \end{array} $$
(8)
This is shown for a number of different networks in Fig. 3.
In a ring of length N the expected route volume 〈V〉_{ring}(n) for a stop list of length n is given by the recursive relation
$$\begin{array}{*{20}l} \langle{V}\rangle(n+1) = \left\{\begin{array}{ll} \frac{N}{4} + \frac{\langle{V}\rangle(n)}{2} + \frac{\langle{V}\rangle(n)^{2}}{3N} & \text{ if} \langle{V}\rangle(n) \leq \frac{N}{2} \\ \frac{N^{2}}{4\langle{V}\rangle(n)} + \frac{5N}{4}  \frac{2\langle{V}\rangle(n)}{3} + \frac{2\langle{V}\rangle(n)^{2}}{3N}& \text{ if} \langle{V}\rangle(n) > \frac{N}{2}. \end{array}\right. \end{array} $$
(9)
A detailed derivation (22) is given in the Appendix A. This approximation holds very well as shown in Fig. 2a.
In a star with N nodes, the number of nodes on the route is approximately equal to the number of unique random draws. This is given by
$$ \langle{V}\rangle_{star}(n)\approx N \left(1\left(\frac{N1}{N}\right)^{n}\right). $$
(10)
This approximation does not account for the special role of the center point of the star, which is always on the route, as soon as the stoplist contains two or more nodes. Nonetheless it holds reasonably well, as shown in Fig. 2b.