A queueing model of dynamic pricing and dispatch control for ride-hailing systems incorporating travel times

A system manager makes dynamic pricing and dispatch control decisions in a queueing network model motivated by ride hailing applications. A novel feature of the model is that it incorporates travel times. Unfortunately, this renders the exact analysis of the problem intractable. Therefore, we study this problem in the heavy traffic regime. Under the assumptions of complete resource pooling and common travel time and routing distributions, we solve the problem in closed form by analyzing the corresponding Bellman equation. Using this solution, we propose a policy for the queueing system and illustrate its effectiveness in a simulation study.

1. The customer demand rate in region i, \(\lambda _i(t)\), depends only on the price \(p_i(t)\).

2. One can assume \(c_i \ge p_i^* = \Lambda _i^{-1}(\lambda _i^*)\) naturally, where \(\lambda ^*\) is defined in Eq. (23) below.

3. In particular, for all j, \(\mu _j^* = \sum _{i=1}^{I}\lambda _i^* A_{ij}\). This is true because there exists only one i such that \(A_{ij} \ne 0\) for each \(j=1,\dots , J\). That is, an activity only uses one server. In matrix notation, \(\mu ^* = A^{\prime }\lambda ^*\), where \(A^{\prime }\) is the transpose of A.

4. Based on intuition from the classical \(M/M/\infty \) queue, this condition implies that the steady-state fraction of jobs in the infinite-server node under the nominal processing plan is equal to one as the number of jobs in the system grows, i.e. as \(n\rightarrow \infty \).

5. The first equality in (35) is proved by applying (34) and noting that \(\left( \Lambda ^n\right) ^{-1}(nx) = \Lambda ^{-1}(x)\) for \(x\in {\mathcal {L}}\). The second equality in (35) then follows by (7).

6. For simplicity, we use the preliminary results from Ata et al. [8] to estimate \(\lambda ^n\) and q (based on a four-year dataset from January 2010 to December 2013). In doing so, we focus on the day shift of the non-holiday weekdays.

7. The estimated performance and the corresponding confidence interval for the sensitivity analysis is also based on 10 macro-replications where each replication has a run-length of 1000 h (and the statistics of the first 200 h are discarded as a warm-up period).

8. In particular, the remainder term is given by

   $$\begin{aligned} R_{\lambda ^*, 3}\left( \frac{1}{\sqrt{n}}\zeta ^n(s)\right) =\sum _{\begin{array}{c} \alpha _1,\dots , \alpha _I\in \left\{ 0,1,2,3\right\} \\ \text {s.t. }\alpha _1+\cdots + \alpha _I = 3 \end{array}} \frac{\partial ^3 \pi \left( \lambda ^* + \frac{C}{\sqrt{n}}\zeta ^n(s)\right) }{\partial x_1^{\alpha _1}\partial x_2^{\alpha _2}\cdots \partial x_I^{\alpha _I}} \prod _{i=1}^{I}\frac{\left( \frac{1}{\sqrt{n}}\zeta _i^n(s)\right) ^{\alpha _i}}{\alpha _i!}\quad \text {for some}\quad C\in (0,1). \end{aligned}$$

Authors and Affiliations

1. The University of Chicago Booth School of Business, Chicago, USA

   Amir A. Alwan, Baris Ata & Yuwei Zhou

Corresponding author

Correspondence to Yuwei Zhou.

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Appendices

Derivations

1.1 Formal derivation of the Brownian control problem

This section provides a formal derivation of the approximating Brownian control problem introduced in Sect. 4. We do not provide a rigorous weak convergence limit theorem. However, the arguments given in support of the approximation can be viewed as a broad outline for such a proof; see Harrison [49, 52, 53] for similar derivations.

We consider a sequence of systems indexed by the system parameter n under the heavy traffic assumption. Then we center the various processes by their mean, scale them appropriately by the system parameter n, and finally pass to the limit as \(n\rightarrow \infty \) formally. To that end, we first define the following (diffusion) scaled processes:

where \(\lfloor x\rfloor \) is the greatest integer less than or equal to x. We also define the following (fluid) scaled processes:

By Donsker’s theorem, the functional central limit theorem for renewal processes, and independence of the stochastic primitives, the processes \({\hat{\Psi }}_i^n\) and \({\hat{N}}_j^n\) converge weakly to independent standard Brownian motions, see Billingsley [28].

As observed in Kogan and Lipster [66], under the heavy traffic assumption, we expect that the number of jobs in the infinite-server node will be n to a first-order approximation. That is, we expect that \({\bar{Q}}^n_0(t)\approx 1\) for \(t\ge 0\) as n gets large. Similarly, we expect the queue lengths at buffers \(1,\dots , I\) to be of order \(\sqrt{n}\). As such, we expect the prices, or equivalently, the demand rates, to deviate from their nominal values only in the second order. That is, we expect \(\lambda _i^n-\lambda ^*_i n = O\left( \sqrt{n}\right) \). Because the demand rates determine the service rates (see Eq. (9)), we expect that \({\bar{\mu }}_j^n(t)\approx \mu _j^*\) for \(t\ge 0\) as n gets large.

By combining Eqs. (145) and (149) with Eqs. (38) and (44), it is straightforward to derive the following scaled system dynamics equations for \(i=1,\dots , I\):

where the second equality holds by Assumption 3 and where the process \(B_i^n\) is given by

Assuming that \(Z_i^n(0)\approx Z_i(0)\) for large n, it is also straightforward to argue that \(B_i^n\) can be approximated by a Brownian motion \(B_i\) with starting state \(Z_i(0)\) that has drift parameter \(\gamma _i={\hat{\eta }}q_i\) and variance parameter

Furthermore, the covariance between the limiting Brownian motion processes is given by

Therefore, replacing \(Z^n\), \(Y^n\), and \(\kappa ^n\), by their formal limits Z, Y, and \(\kappa \), we arrive at the following system dynamics equations in the approximating Brownian control problem for \(i=1,\dots , I\):

Equations (19) and (39) of the system state also imply that \(Z_0^n(t) = -\sum _{i=1}^{I}Z_i^n(t)\) and that \(Z_i^n(t)\ge 0\) for \(i=1,\dots , I\) and \(t\ge 0\). Thus, in the approximating BCP, the following relationships hold for \(t\ge 0\):

Similarly, it is clear that Eqs. (9) and (43) and (44) give rise to Eq. (52) in the BCP; Eqs. (17) and (41) give rise to Eq. (54); and Eq. (42) gives rise to Eq. (51).

To complete the formal derivation of the Brownian control problem, we argue that \({\hat{V}}^n\approx \xi \) for large n, where \({\hat{V}}^n\) and \(\xi \) are given by Eqs. (46) and (47), respectively. First, observe that by Taylor’s theorem we have

where \(R_{\lambda ^*, 3}\left( \frac{1}{\sqrt{n}}\zeta ^n(s)\right) =O(n^{-3/2})\) is a third-order remainder term.^{Footnote 8} Moreover, note that the term \(\nabla \pi \left( \lambda ^*\right) ^{\prime }\zeta ^n(s)/\sqrt{n}\) vanishes because \(\lambda ^*\) is a maximizer of \(\pi \left( \lambda \right) \) and is in the interior of the feasible region \({\mathcal {L}}\) (see Assumption 2), implying that \(\nabla \pi \left( \lambda ^*\right) =0\). Therefore, we have that

where \(H=-\frac{1}{2}\nabla ^2\pi \left( \lambda ^*\right) \). Using this and Eqs. (35) and (43), it follows that

Finally, using Eqs. (39), (41) and (46), and (150), it is straightforward to derive the following:

Therefore, replacing \({\hat{V}}^n\), \(Z^n\), \(\zeta ^n\), and \(U^n\) by their formal limits \(\xi \), Z, \(\zeta \), and U, we arrive at the following cost process of the approximating Brownian control problem:

Note that it is a diagonal matrix, i.e., \(H=\textrm{diag}(\alpha _1,\ldots ,\alpha _I)\) where

Using this we further simplify the limiting cost process \(\xi (t)\) as follows:

1.2 Derivation of Eq. (143)

Recall that the proposed chosen demand rates are

where \(q(t) = \frac{1}{2\alpha _i}v\left( \frac{W^n(t)}{\sqrt{n}}\right) \). Therefore, the proposed pricing policy for region i is given as follows:

where the third equality follows from the fact that \(\left( \Lambda _i^n\right) ^{-1}(nx) = \Lambda _i^{-1}(x)\) for \(x\in {\mathcal {L}}\). Then note that by Taylor’s theorem we have

which implies that

As an aside, observe that by rearranging terms we have

This implies that our proposed dynamic pricing policy coincides with the static prices to a first-order approximation, but deviates from the static prices on the second order, i.e., order \(1/\sqrt{n}\).

Miscellaneous proofs

Proof of Lemma 1

This proof follows in an almost identical fashion to Lemma 2 in Ata et al. [9], but we include it for completeness. It consists of four steps. We let \(e^n\) denote the nth unit basis vector in a Euclidean space of appropriate dimension. That is, the nth component of \(e^n\) is one, whereas its other components are zero. Moreover, recall from the discuss following Assumption 3 that for a vector \(y\in {\mathbb {R}}^J\), we write \(y = \left( y_B, y_N\right) \) where \(y_B\in {\mathbb {R}}^b\) and \(y_N\in {\mathbb {R}}^{J-b}\).

Step 1: Consider the set of basic activity rates that do not cause any server idleness, i.e., \(\left\{ y\in {\mathbb {R}}^J: By_B=0,\,y_N=0\right\} \). First, we show that this set is the span of \(\bar{\bar{C}}\), defined next:

where \({\bar{C}}=\left\{ \left( j,j^{\prime }\right) : j,\,j^{\prime }\in \left\{ 1,\dots , b\right\} \text { such that }s(j) = s(j^{\prime })\right\} \). That is, \({\bar{C}}\) is the set of all pairs of basic activities undertaken by the same server. Note that the difference \(e^j - e^{j^{\prime }}\) in Eq. (151) captures the trade-off server s(j) makes by increasing the rate at which activity j is undertaken (from its nominal value \(x_j^*\)) at the expense of decreasing the rate of activity \(j^{\prime }\). By making such adjustments to the nominal basic activity rates \(x_B^*\), the system manager can redistribute the workload between buffers b(j) and \(b(j^{\prime })\) without incurring any idleness. As such, we intuitively expect that taking linear combinations of such activity rates in \(\bar{\bar{C}}\) should yield the set of activity rates that do not result in any idleness, i.e., the set \(\left\{ y\in {\mathbb {R}}^J: By_B=0,\,y_N=0\right\} \). In summary, in Step 1 we prove that

To prove this, we show inclusions of both sets. First, let \(y\in \Bigg \{y\in {\mathbb {R}}^J: By_B= 0,y_N =0\Bigg \}\). To prove that \(y\in \text {span}\left( \bar{\bar{C}}\right) \), we show that there exist constants \(a_{jj^{\prime }}\), \((j,j^{\prime })\in {\bar{C}}\), such that \(y = \sum _{(j,j^{\prime })\in {\bar{C}}}a_{jj^{\prime }}\left( e^{j}- e^{j^{\prime }}\right) .\) To find these constants, it will be convenient to define the sets

where \({\mathcal {A}}_i\) is the set of activities undertaken by server i, see Eq. (3). To be more specific, \(\bar{{\mathcal {A}}}_i\) consists of all basic activities undertaken by server i. After possibly relabeling, suppose that the basic activities are ordered so that

where \(0=b_0<b_1<b_2<\cdots < b_{I}=b.\) We define constants \(a_{jj^{\prime }}\) for \((j,j^{\prime })\in {\bar{C}}\) as follows:

Therefore, we have that

the first two equalities following from the definition of the \(a_{jj^{\prime }}\),

the fourth equality from algebraic rearrangements, and the fifth equality from canceling terms. To derive the sixth equality note that y satisfies \(By_B=0\), which implies

Equivalently, we have that

Substituting this for the last term of the fifth equality yields the sixth equality. Finally, the eighth equality from the facts that \(y_N=0\) and that the sets \(\bar{{\mathcal {A}}}_i\), \(i=1,\dots , I\), partition the basic activities. Since \(y = \sum _{j=1}^{J}y_j e^j\), we conclude that \(y\in \text {span}\left( \bar{\bar{C}}\right) \).

Conversely, let \(y\in \text {span}\left( \bar{\bar{C}}\right) \). Then there are constants \(a_{jj^{\prime }}\), \((j,j^{\prime })\in {\bar{C}}\), such that

Since \({\bar{C}}\) consists only of pairs of basic activities, it follows that \(y_N = 0\). Furthermore, for \((j,j^{\prime })\in {\bar{C}}\) and \(i\in \left\{ 1,\dots , I\right\} \), we have

the second equality holding by Eq. (1) and the fourth equality holding since \(s(j) = s(j^{\prime })\). Therefore, \(A\left( e^j - e^{j^{\prime }}\right) =0\) for all \((j,j^{\prime })\in {\bar{C}}\), implying that \(Ay = 0\) by linearity. So, \(y\in \left\{ y\in {\mathbb {R}}^J: Ay= 0,\,y_N =0\right\} \).

Step 2: In this step, we show that \({\mathcal {N}} = \text {span}\left( {\tilde{C}}\right) \), where \({\tilde{C}} = \left\{ Ry: y\in \bar{\bar{C}}\right\} .\) To see this, recall that \({\mathcal {N}} = \left\{ Hy_B: By_B=0,\, y_B\in {\mathbb {R}}^b\right\} = \left\{ Ry: Ay=0,\, y_N=0\right\} .\) Thus, it follows from Step 1 and the definition of \({\tilde{C}}\) that \({\mathcal {N}} = \text {span}\left( {\tilde{C}}\right) \).

Step 3: In this step, we show that \({\tilde{C}} = \Bigg \{\mu _j^*\left( e^{b(j)} - e^{b(j^{\prime })}\right) : \left( j,j^{\prime }\right) \in {\bar{C}},\, e^{b(i)},\,e^{b(j^{\prime })}\in {\mathbb {R}}^I\Bigg \}\). To see this, note that for \((j,j^{\prime })\in {\bar{C}}\) and \(i\in \left\{ 1,\dots , I\right\} \), we have that

the second equality following from Eqs. (2) and (107) and the fourth equality following from the fact that \(s(j) = s(j^{\prime })\) (since \((j,j^{\prime })\in {\bar{C}}\)) and Eq. (24). That is,

Then using the definition of \(\bar{\bar{C}}\), we write

where the last equality follows from Eq. (152). Hence, the result holds. In particular, by the definition of buffer communication, note that

Step 4: We consider the matrix M defined in Lemma 1 (see Eq. (59)) and show that its rows form a basis for \({\mathcal {M}}\). To that end, let \(M^l\), \(l=1,\dots , L\), be the rows of the matrix M given in Eq. (59). Since the buffer pools partition the servers, the rows of M are linearly independent. Thus, to complete the proof, it suffices to show that \({\mathcal {M}}=\text {span}\left( M^1,\dots , M^L\right) \). Recalling that \({\mathcal {M}}= {\mathcal {N}}^{\perp }\) and \({\mathcal {N}} = \text {span}\left( {\tilde{C}}\right) \), it follows that \(a\in {\mathcal {M}}\) if and only if \(a\cdot z = 0\) for all \(z\in {\tilde{C}}\). Moreover, since \(\mu _j^*>0\) for all \(j\in \left\{ 1,\dots , b\right\} \), it follows from Step 3 that

Therefore, \(a\in {\mathcal {M}}\) if and only if \(a_{i} = a_{i^{\prime }}\) for all buffers i and \(i^{\prime }\) that communicate directly.

To prove that \({\mathcal {M}}=\text {span}\left( M^1,\dots , M^L\right) \) we show inclusions of both sets. On the one hand, let \(a\in {\mathcal {M}}\). Then \(a_i = a_{i^{\prime }}\) for all buffers i and \(i^{\prime }\) that communicate directly. By definition of buffer communication, it immediately follows that \(a_i = a_{i^{\prime }}\) for all buffers i and \(i^{\prime }\) that communicate. That is, \(a_i = a_{i^{\prime }}\) for all buffers i and \(i^{\prime }\) that are in the same buffer pool. Thus, \(a\in \text {span}\left( M^1,\dots , M^L\right) \), implying that \({\mathcal {M}}\subseteq \text {span}\left( M^1,\dots , M^L\right) \). On the other hand, to show that \(\text {span}\left( M^1,\dots , M^L\right) \subseteq {\mathcal {M}}\), it suffices to show that \(M^l\in {\mathcal {M}}\) for each \(l=1,\dots , L\). To that end, it is enough to show that \(M^l_{i} = M^l_{i^{\prime }}\) for all buffers i and \(i^{\prime }\) that communicate directly. However, this trivially holds by Eq. (59), since buffers i and \(i^{\prime }\) that communicate directly are in the same buffer pool. Thus, \(\text {span}\left( M^1,\dots , M^L\right) \subseteq {\mathcal {M}}.\) \(\square \)

Proof of Lemma 2

It is enough to show that \((MR)_{lj}=(GA)_{lj}\) for all \(l=1,\dots , L\) and \(j=1,\dots , J\), where G is given by Eq. (60). Indeed, by Eqs. (122), (116), and (59),

On the other hand, by Eqs. (120) and (60),

Note that by Eq. (24) we have \(\mu _j^*=\lambda ^*_{s(j)}\) and by Eq. (58) we have that \(b(j)\in {\mathcal {P}}_l\) if and only if \(s(j)\in {\mathcal {S}}_l\). Thus, the desired result immediately follows by Eqs. (153) and (154). \(\square \)

Proof of Lemma 3

When \(L=1\), all buffers are in a single buffer pool. Thus, it follows immediately from Eq. (59) that \(M= e^{\prime }\). Furthermore, by definition of buffer communication and Eq. (58), there is a single server pool. It then follows from Eq. (60) that \(G=\left( \lambda ^*\right) ^{\prime }\).

To prove the first relationship in Eq. (69), note that

where the second equality follows from \(M=e^{\prime }\) and where the fourth equality follows from the fact that q is a probability vector. To prove the second relationship in Eq. (69), first note that \(MC=e^{\prime }\in {\mathbb {R}}^J\). This follows from \(M=e^{\prime }\in {\mathbb {R}}^I\), the definition of C in Eq. (142), and the fact that C has one nonzero element per column. Therefore,

the fourth equality following from the heavy traffic assumption, see Eq. (29). \(\square \)

Proof of Lemma 4

This is a straightforward convex optimization problem. Forming the Lagrangian

where \(\nu \) is the Lagrange multiplier, the necessary first-order conditions then give

Substituting this into the constant \(e'\zeta =x\) yields \(\nu =2x/{\hat{\alpha }}\) and

The optimality of this solution follows from the convexity of the objective. Substituting (155) in the objective function yields \(c(x)=x^2/{\hat{\alpha }}\) as desired. \(\square \)

Proof of Proposition 1

Let \((Y,\zeta )\) be an admissible control for (48) and (54) with the corresponding state process Z and idleness process U. Letting \(W(t)=MZ(t)\) for \(t\ge 0\), (48) implies that (65) holds, and (64) holds by definition. Similarly, (66) and (67) follow from (53) and (54) whereas (68) follows from (52). Thus, \((Z, U, \zeta )\) of the BCP formulation (48) and (54) is an admissible policy for the RBCP (63) and (68). Because the two formulations have the same process \(Z, U, \zeta \), they have the same cost.

The converse follows exactly as in (the second part of) the proof of Theorem 1 in Harrison and Van Mieghem [56] (see pages 753–754) with the only substantive difference being (aside from the obvious notational differences) the process X on their Eq. (36) on page 755 is replaced with

in our setting. Then following the same steps in their proof shows that the analogy of the process Y (in our setting) defined as in their Eq. (35) and \(\zeta \) is admissible for our BCP (48) and (54). Moreover, because \((Y, \zeta )\) results in the same queue length process Z. Its cost is the same as that of the policy \((Z, U, \zeta )\) for RBCP (63) and (68). \(\square \)

Proof of Proposition 2

Given an admissible policy \(\theta \) for EWF and the corresponding process W, L, we set \(Z_{i^*}\equiv W\) and \(Z_i\equiv 0\) for \(i\not = i^*\) and \(U_{k^*} \equiv L\) and \(U_k\equiv 0\) for \(k\not = k^*\). Moveover, we set \(\zeta _i(s)=\theta (s)/(\alpha _i {\hat{\alpha }}_i)\) for \(i=1,\ldots , I\), which results in \(\sum _{i=1}^I \alpha _i \zeta _i^2 (s) = c(\theta (s))\) by Lemma 4. Then it follows from (77) and (78) that \((Z, U, \zeta )\) has the same cost in RBCP as \(\theta \) does in EWF.

To prove the converse, let \(Z, U, \zeta \) be an admissible policy for RBCP, and let

It is easy to verify that \(\theta (\cdot )\) is admissible for EWF. Moreover, \(c(\theta (s))\le \sum _{i=1}^I \alpha _i \zeta _i^2 (s)\) by Lemma 4 and that \(h W(s) \le \sum _{i=1}^I (h_i-h_0) Z_i(s)\) and \(r L(s)\le c' U(s)\) for \(s\ge 0\). Thus, the cost of \(\theta \) for the EWF is less than or equal to that of the policy \((Z, U, \zeta )\) for the RBCP. \(\square \)

Proof of Proposition 3

Consider the auxiliary stationary reflected diffusion on \([0,\infty )\), denoted by \(\left\{ {\tilde{W}}(t),\,t\ge 0\right\} \), associated with the drift rate function \(-\left( \eta y -a + \theta ^*(y)\right) \) and variance parameter \(\sigma ^2\). As noted on pages 470–471 of Browne and Whitt [34]—also see Mandl [74] and Karlin and Taylor [64]—its probability density function, denoted by \(\varphi \), is given as follows:

provided all integrals are finite, which we verify next. To this end, let \(k=\inf \left\{ y\ge 0: v(y)\ge 0\right\} \) where \((v,\beta ^*)\) solve the Bellman equation (89) and (90), and note from Eq. (90) that \(-r\le v(y)\le 0\) for \(y\le k\) and \(0\le v(y) \le h/\eta \) for \(y\ge k\). In order to verify the integrals above are finite, using Eq. (91) note that

from which we also deduce that the integral in the denominator of the right hand side of Eq. (123) is finite. Moreover, it follows from Eq. (157) that the stationary diffusion \({\tilde{W}}\) has finite moments. In particular,

Next, we define another auxiliary stationary diffusion, denoted by \({\tilde{W}}^*\), as follows:

Noting \(W^*(0)<{\tilde{W}}^*(0)\) almost surely, we define the stopping time \(\tau \) as follows:

and introduce the following process:

By the strong Markov property of diffusions, \({\hat{W}}^*\) has the same distribution as \(W^*\). Moreover,

Therefore, we conclude that

where the second equality follows from the definition of \({\tilde{W}}^*\), the third equality from the stationarity of \({\tilde{W}}\), and the last equality from Eq. (158). Thus, we conclude from \(W^*(t)\ge 0\) for \(t\ge 0\) and Eq. (159) that

as desired. \(\square \)

The next lemma aids in the proof of Lemma 6. To state the result, it will be convenient to rewrite Eqs. (103) and (104) as follows:

where \(q_0(x)=\frac{2}{\sigma ^2}\left( \beta - hx\right) \), \(q_1(x)=\frac{2}{\sigma ^2}(\eta x-a)\), and \( q_2=\frac{{\hat{\sigma }}}{2\sigma ^2}>0\) for \(x\ge 0\).

Lemma 23

For each \(v\in C^1[0,\infty )\) satisfying Eqs. (160) and (161), \(y(x) = \exp \left\{ -q_2\int _{0}^{x}v(t)\,dt\right\} \) satisfies

Conversely, for each \(y\in C^2[0,\infty )\) satisfying Eqs. (162) and (163), \(v=-y^{\prime }/(q_2y)\) satisfies Eqs. (160)–(161).

Proof of Lemma 23

Let \(v\in C^1[0,\infty )\) satisfy Eqs. (160) and (161) and let \(y(x) = \exp \left\{ -q_2\int _{0}^{x}v(t)\,dt\right\} \). Then it follows that

Moreover, \(y(0) = \exp \left\{ -q_2\cdot 0\right\} = 1\) and \(y^{\prime }(0) = -q_2\exp \left\{ -q_2\cdot 0\right\} v(0) = rq_2\).

On the other hand, let \(y\in C^2[0,\infty )\) satisfy Eqs. (162) and (163) and let \(v=-y^{\prime }/(q_2y)\). Then it follows that

Moreover, \(v(0) = -y^{\prime }(0)/\left( q_2y(0)\right) = -\left( rq_2\right) /\left( q_2\cdot 1\right) =-r\). This completes the proof.\(\square \)

Proof of Lemma 6

It is known that Eqs. (162) and (163) can be transformed into a degenerate hypergeometric equation known as a Kummer’s equation; see Polyanin and Zaitsev [79]. Such equations are known to have confluent hypergeometric function solutions; see Bateman and Erdélyi [22] and Abramowitz and Stegun [1]. It then follows from Lemma 23 that Eqs. (160) and (161) have a solution v. To complete the proof, we must show that the solution v to Eqs. (160) and (161) is unique. To this end, define the function f by

To prove uniqueness, it is enough to show that f is locally Lipschitz in v, i.e., that f is Lipschitz in v when restricted to the compact domain \([0,N]\times [-M, M]\) where \(N,M>0\). More specifically, local Lipschitzness will demonstrate uniqueness on each compact interval, which can then be easily extended to the positive real line. To this end, for \(x\in [0,N]\) and \(v_1,v_2\in [-M,M]\) we have that

Thus, f is locally Lipschitz in v. This completes the proof.\(\square \)

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