Probabilistic spatio-temporal resource search

Abstract

In this paper, we deal with the resource search problem in a probabilistic setting. In a resource search problem, there are spatially located static resources and a mobile agent. The agent looks to obtain one of the resources while minimizing the cost. This cost may consist of different types of costs the agent has to pay, from travel time to the cost of obtaining a certain resource. We assume that the agent has no knowledge of exact availability of the resources in real-time, but some prior or partial data gives estimations of this information. This model applies to many situations that arise in urban transportation systems, such as drivers looking for street parking, taxis looking for new customers, and electric vehicles looking for charging stations. Our approach to the resource search problem only employs uncertain information about resource availability, minimizes the expected cost, and utilizes concepts from decision theory. A simulation that uses real-world data is used to compare our approach to alternatives.

Appendix:: Proofs

Proof Proof for Theorem 1

First, we define some notation. Let $({C_{i}^{k}})'$ denote the expect cost (not necessarily the minimum) of any possible path with k-step look-ahead from node v _i . Let $(C_{ij}^{k})'$ denote the expected cost (not necessarily the minimum) of any possible path with k-step look-ahead given that v _i and v _j are the first two nodes of the path.

When k=0, $\forall i = 1, \dots , n$, ${C_{i}^{0}} = \beta _{i}$ is the only possible expected cost for the zero-length path starting from v _i . Therefore ${C_{i}^{0}}$ is the minimum.

Assume that when k = k ₀−1, $\forall i = 1, \dots , n$, $C^{k_{0}-1}_{i}$ is the minimum expected general cost for all paths starting at v _i with length k ₀−1 or less. That is, C i k ₀−1≤(C i k ₀−1)^′.

Then, when k = k ₀, for any given v _i , we consider two categories of possible successor nodes for v _i ($\{v_{j^{\prime }}\}$ and $\{v_{j^{\prime \prime }}\}$):

• 1) $\forall j^{\prime }$ such that $e_{ij^{\prime }}$ exists and $\mathit {uc}_{ij^{\prime }} \leq C^{k_{0}-1}_{j^{\prime }}$.

By Eq. 3,

$$C^{k_{0}}_{ij^{\prime}} = \mathit{tc}_{ij^{\prime}} + p_{ij^{\prime}} \ \mathit{uc}_{ij^{\prime}} + (1-p_{ij^{\prime}})C^{k_{0}-1}_{j^{\prime}} .$$

Because $\mathit {uc}_{ij^{\prime }} \leq C^{k_{0}-1}_{j^{\prime }} \leq (C^{k_{0}-1}_{j^{\prime }})'$, by Eq. 21,

$$(C^{k_{0}}_{ij^{\prime}})' = \mathit{tc}_{ij^{\prime}} + p_{ij^{\prime}} \ \mathit{uc}_{ij^{\prime}} + (1-p_{ij^{\prime}})(C^{k_{0}-1}_{j^{\prime}})' .$$

Clearly, $C^{k_{0}}_{ij^{\prime }} \leq (C^{k_{0}}_{ij^{\prime }})'$.

• 2) $\forall j^{\prime \prime }$ such that $e_{ij^{\prime \prime }}$ exists and $\mathit {uc}_{ij^{\prime \prime }} > C^{k_{0}-1}_{j^{\prime \prime }}$.

By Eq. 3,

$$C^{k_{0}}_{ij^{\prime\prime}} = \mathit{tc}_{ij^{\prime\prime}} + C^{k_{0}-1}_{j^{\prime\prime}} .$$

The relationship between $\mathit {uc}_{ij^{\prime \prime }}$ and $(C^{k_{0}-1}_{j^{\prime \prime }})'$ in this second case can be either way.

• a) When $\mathit {uc}_{ij^{\prime \prime }} \leq (C^{k_{0}-1}_{j^{\prime \prime }})'$, by Eq. 21,

  $$(C^{k_{0}}_{ij^{\prime\prime}})' = \mathit{tc}_{ij^{\prime\prime}} + p_{ij^{\prime\prime}} \ \mathit{uc}_{ij^{\prime\prime}} + (1-p_{ij^{\prime}})(C^{k_{0}-1}_{j^{\prime\prime}})' .$$

In this case, $C^{k_{0}}_{ij^{\prime \prime }} \leq (C^{k_{0}}_{ij^{\prime \prime }})'$ because $C^{k_{0}-1}_{j^{\prime \prime }} < \mathit {uc}_{ij^{\prime \prime }}$ and $C^{k_{0}-1}_{j^{\prime \prime }} \leq (C^{k_{0}-1}_{j^{\prime \prime }})'$.

• b) When $\mathit {uc}_{ij^{\prime \prime }} > (C^{k_{0}-1}_{j^{\prime \prime }})'$, by Eq. 21,

  $$(C^{k_{0}}_{ij^{\prime\prime}})' = \mathit{tc}_{ij^{\prime\prime}} + (C^{k_{0}-1}_{j^{\prime\prime}})' .$$

In this case, again $C^{k_{0}}_{ij^{\prime \prime }} \leq (C^{k_{0}}_{ij^{\prime \prime }})'$.

Integrating cases 1) and 2), we can conclude that ∀j such that e _{i j} exists, C i j k ₀≤(C i j k ₀)^′. Therefore, by Eq. 2, ∀j such that e _{i j} exists, $C^{k_{0}}_{i} \leq \min\limits \left \{ C^{k_{0}}_{ij}, \beta _{i} \right \} \leq \min\limits \left \{ (C^{k_{0}}_{ij})', \beta _{i}\right \}$. Because {v _j } here covers all possible successors of v _i , we can get C i k ₀≤(C i k ₀)^′.

By induction, we can reach the conclusion that ${C^{k}_{i}}$ is the minimum expected general cost for all paths starting at v _i with length K or less, in other words, ${C_{i}^{K}} \leq ({C_{i}^{K}})'$.

The time complexity O(n d K) is a direct result of applying Eqs. 2 and 3 for dynamic programming (refer to Algorithm 1). □

Proof Proof for Theorem 2

When k=0, ∀i,j such that e _{i j} exists, by Eq. 2, $\mathit {uc}_{ij} \leq \beta _{j} = {C^{0}_{j}}$.

Assume that when k = k ₀−1, ∀i,j such that e _{i j} exists, it holds that u c _{i j} ≤C j k ₀−1, then for k = k ₀ and any given i, ∀j such that e _{i j} exists, by Eq. 3:

$$\begin{array}{@{}rcl@{}} C^{k_{0}}_{ij} &= & \mathit{tc}_{ij} + p_{ij} \ \mathit{uc}_{ij} + (1-p_{ij})C^{k_{0}-1}_{j} \\ &\geq & \mathit{tc}_{ij} + p_{ij} \ \mathit{uc}_{ij} + (1-p_{ij})\mathit{uc}_{ij}\\ &\geq & \mathit{uc}_{ij} \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} C^{k_{0}}_{i} &= & \displaystyle\min\limits_{\forall j, \text{ s.t.\ } e_{ij} \text{ exists}}\left\{C^{k_{0}}_{ij}, \ \beta_{i} \right\} \\ &\geq & \mathit{uc}_{ij} \end{array} $$

This can be rewritten as u c _{i j} = u c _{j i} ≤C j k ₀.

By induction, we can get the conclusion in the theorem. □

The following lemma is necessary to prove Lemma 1:

Lemma 5

Let {a _i }, {b _i }, $i = 1,2,\dots ,n$ be two real sequences with the same length n, then

$$\displaystyle\left|\min\limits_{1\leq i \leq n} \{a_{i}\} - \min\limits_{1\leq i \leq n} \{b_{i}\}\right| \leq \max\limits_{1\leq i \leq n} \{|a_{i}-b_{i}|\}.$$

Proof

$$\begin{array}{@{}rcl@{}} \min\limits_{1 \leq i \leq n}\{a_{i}\} - \min\limits_{1 \leq i \leq n}\{b_{i}\} &= & \max\limits_{1 \leq i \leq n}\left\{\min\limits_{1 \leq j \leq n}\{a_{j}\} - b_{i} \right\} \\ &\leq & \max\limits_{1 \leq i \leq n}\{a_{i} - b_{i}\} \\ &\leq & \max\limits_{1\leq i \leq n} \{|a_{i}-b_{i}|\} . \end{array} $$

$$\begin{array}{@{}rcl@{}} \min\limits_{1 \leq i \leq n}\{a_{i}\} - \min\limits_{1 \leq i \leq n}\{b_{i}\} &= & \min\limits_{1 \leq i \leq n}\left\{a_{i} - \min\limits_{1 \leq j \leq n}\{b_{j}\} \right\} \\ &\geq & \min\limits_{1 \leq i \leq n}\{a_{i} - b_{i}\} \\ &= & - \max\limits_{1\leq i \leq n} \{-(a_{i}-b_{i})\} \\ &\geq & - \max\limits_{1\leq i \leq n} \{|a_{i}-b_{i}|\} . \end{array} $$

From these, we can get the inequality in the lemma. □

Proof Proof for Lemma 1

Firstly, we prove that $\forall 1\leq i \leq n, |C^{k+1}_{i} - C^{k^{\prime }+1}_{i}| \leq \gamma ||\boldsymbol {C^{k}}-\boldsymbol {C^{k^{\prime }}}||$ using Lemma 5 and Eqs. 2 and 11:

$$\begin{array}{@{}rcl@{}} && |C^{k+1}_{i} - C^{k^{\prime}+1}_{i}| \\ &&\qquad = \left|\min\limits_{\forall j, \text{ s.t.\ } e_{ij} \text{ exists}}\left\{\mathit{tc}_{ij} + p_{ij} \ \mathit{uc}_{ij}+(1-p_{ij}){C^{k}_{j}}, \ \beta_{i} \right\} \right.\\ &&\qquad\quad \left. -\min\limits_{\forall j, \text{ s.t.\ } e_{ij} \text{ exists}}\left\{\mathit{tc}_{ij} + p_{ij} \ \mathit{uc}_{ij}+(1-p_{ij})C^{k^{\prime}}_{j}, \ \beta_{i} \right\}\right| \\ &&\qquad \leq \max\limits_{\forall j, \text{ s.t.\ } e_{ij} \text{ exists}} \left\{\left| (\mathit{tc}_{ij} + p_{ij} \ \mathit{uc}_{ij}+(1-p_{ij}){C^{k}_{j}}) \vphantom{C^{k^{\prime}}_{j}} \right.\right. \\ &&\qquad\quad \left.\left. -(\mathit{tc}_{ij} + p_{ij} \ \mathit{uc}_{ij}+(1-p_{ij})C^{k^{\prime}}_{j}) \right|, \ 0 \right\} \\ &&\qquad = \max\limits_{\forall j, \text{ s.t.\ } e_{ij} \text{ exists}} \left\{ (1-p_{ij}) \left|{C^{k}_{j}}-C^{k^{\prime}}_{j} \right| \right\} \\ && \qquad \leq \max\limits_{\forall j, \text{ s.t.\ } e_{ij} \text{ exists}} \left\{ (1-p_{\mathit{min}}) \left|{C^{k}_{j}}-C^{k^{\prime}}_{j} \right| \right\} \\ && \qquad\leq (1-p_{\mathit{min}}) \max\limits_{1\leq j \leq n} \left\{\left|{C^{k}_{j}}-C^{k^{\prime}}_{j} \right|\right\} \\ &&\qquad = \gamma ||\boldsymbol{C^{k}}-\boldsymbol{C^{k^{\prime}}}|| . \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} &&||\boldsymbol{C^{k+1}}-\boldsymbol{C^{k^{\prime}+1}}|| \\ && \qquad= \max\limits_{1\leq i \leq n} \left\{\left|C^{k+1}_{i} - C^{k^{\prime}+1}_{i} \right|\right\} \\ && \qquad \leq \gamma ||\boldsymbol{C^{k}}-\boldsymbol{C^{k^{\prime}}}|| . \end{array} $$

□

Proof Proof for Lemma 2

Note that because ∀i,j such that e _{i j} exists, u c _{i j} = u c _{j i} ≤β _i , by Theorem 2, Eq. 21 only presents its first case. By expanding the recursion in Eq. 21, the expected cost of this path is¹²

$$\begin{array}{@{}rcl@{}} &&{} C_{\{v_{0} \rightarrow v_{1} \rightarrow \cdots\} }\\ && = {\sum}_{k=1}^{\infty}{\left( \left( \mathit{tc}_{{k-1}, {k}} + \mathit{uc}_{{k-1}, {k}} \cdot p_{{k-1}, {k}}\right) {\prod}_{j=1}^{k-1}{\left( 1-p_{{j-1}, {j}}\right)} \right)} + \lim_{k \to \infty}{\left( \beta_{{k}} {\prod}_{j=1}^{k}{\left( 1-p_{{j-1}, {j}}\right)} \right)} \\ && \leq {\sum}_{k=1}^{\infty}{\left( \left( \mathit{tc}_{\mathit{max}} + \mathit{uc}_{\mathit{max}} \cdot p_{\mathit{max}}\right) \left( 1-p_{\mathit{min}}\right)^{k-1} \right)} + \beta_{\mathit{max}} \lim_{k \to \infty}{\left( 1-p_{\mathit{min}}\right)^{k}}\\ && =\ \frac{\mathit{tc}_{\mathit{max}} + \mathit{uc}_{\mathit{max}} \cdot p_{\mathit{max}} }{p_{\mathit{min}}} . \end{array} $$

□

Proof Proof for Lemma 4

Using Lemma 1 and the triangle inequality of max norm, we get

$$\begin{array}{@{}rcl@{}} && ||\boldsymbol{C^{k+1}}-\boldsymbol{C}|| \\ &&\qquad \leq (1-p_{\mathit{min}}) ||\boldsymbol{C^{k}}-\boldsymbol{C}||\\ &&\qquad \leq (1-p_{\mathit{min}}) ||\boldsymbol{C^{k+1}}-\boldsymbol{C^{k}}|| + (1-p_{\mathit{min}}) ||\boldsymbol{C^{k+1}}-\boldsymbol{C}|| \\ &&\qquad \leq \epsilon_{0} \ p_{\mathit{min}} + (1-p_{\mathit{min}}) ||\boldsymbol{C^{k+1}}-\boldsymbol{C}||, \end{array} $$

which implies ||C ^k+1−C||≤𝜖 ₀. □