Maximum-stability dispatch policy for shared autonomous vehicles - PowerPoint PPT Presentation

Maximum-stability dispatch policy for shared autonomous vehicles Michael W. Levin, Di Kang

Shared autonomous vehicles (SAVs) SAV service currently in testing on public roads SAVs have safety driver Motivation Max-stability dispatch for SAVs M. W. Levin, D. Kang 2

Motivation Max-stability dispatch for SAVs M. W. Levin, D. Kang 3

SAV : personal vehicle replacement rates 1 SAV : 10 personal vehicles a 1 SAV : 9 personal vehicles b 1 SAV : 3 personal vehicles c a Daniel J Fagnant and Kara M Kockelman. “The travel and environmental implications of shared autonomous vehicles, using agent-based model scenarios”. In: Transportation Research Part C: Emerging Technologies 40 (2014), pp. 1–13. b Daniel J Fagnant, Kara M Kockelman, and Prateek Bansal. “Operations of Shared Autonomous Vehicle Fleet for Austin, Texas Market”. In: Transportation Research Record: Journal of the Transportation Research Board 2536 (2015), pp. 98–106. c Kevin Spieser et al. “Toward a systematic approach to the design and evaluation of automated mobility-on-demand systems: A case study in Singapore”. In: Road Vehicle Automation . NY: Springer, 2014, pp. 229–245. Motivation Max-stability dispatch for SAVs M. W. Levin, D. Kang 3

Agent-based simulation 1 1 Daniel J Fagnant and Kara M Kockelman. “Dynamic ride-sharing and fleet sizing for a system of shared autonomous vehicles in Austin, Texas”. In: Transportation 45.1 (2018), pp. 143–158. Motivation Max-stability dispatch for SAVs M. W. Levin, D. Kang 4

Agent-based simulation 2 2 T Donna Chen, Kara M Kockelman, and Josiah P Hanna. “Operations of a shared, autonomous, electric vehicle fleet: Implications of vehicle & charging infrastructure decisions”. In: Transportation Research Part A: Policy and Practice 94 (2016), pp. 243–254. Motivation Max-stability dispatch for SAVs M. W. Levin, D. Kang 5

SAV : personal vehicle replacement rates 1 SAV : 10 personal vehicles a 1 SAV : 9 personal vehicles b 1 SAV : 3 personal vehicles c a Daniel J Fagnant and Kara M Kockelman. “The travel and environmental implications of shared autonomous vehicles, using agent-based model scenarios”. In: Transportation Research Part C: Emerging Technologies 40 (2014), pp. 1–13. b Daniel J Fagnant, Kara M Kockelman, and Prateek Bansal. “Operations of Shared Autonomous Vehicle Fleet for Austin, Texas Market”. In: Transportation Research Record: Journal of the Transportation Research Board 2536 (2015), pp. 98–106. c Kevin Spieser et al. “Toward a systematic approach to the design and evaluation of automated mobility-on-demand systems: A case study in Singapore”. In: Road Vehicle Automation . NY: Springer, 2014, pp. 229–245. Motivation Max-stability dispatch for SAVs M. W. Levin, D. Kang 6

Queueing model

Passenger queueing model Define a queue of waiting passengers at each zone: w rs ( t ) . Conservation of waiting passengers:     w rs ( t + 1) = w rs ( t ) + d rs ( t ) − min � y rs rj ( t ) , w rs ( t )   j ∈A where d rs ( t ) are random variables with mean ¯ d rs . y rs rj ( t ) ≤ p r ( t ) is vehicles departing r for s to link j Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 8

Passenger queueing model Define a queue of waiting passengers at each zone: w rs ( t ) . Conservation of waiting passengers:     w rs ( t + 1) = w rs ( t ) + d rs ( t ) − min � y rs rj ( t ) , w rs ( t )   j ∈A where d rs ( t ) are random variables with mean ¯ d rs . y rs rj ( t ) ≤ p r ( t ) is vehicles departing r for s to link j This defines a Markov chain on the state space N |Z| 2 . Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 8

Vehicle queueing model x rs j ( t ) is the number of vehicles on link j traveling from r to s p r ( t ) is the number of vehicles parked at r � � x rs � j ( t ) + p r ( t ) = F j ∈A ( r,z ) ∈Z 2 r ∈Z Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 9

Vehicle queueing model x rs j ( t ) is the number of vehicles on link j traveling from r to s p r ( t ) is the number of vehicles parked at r � � x rs � j ( t ) + p r ( t ) = F j ∈A ( r,z ) ∈Z 2 r ∈Z Conservation of link queues: � � � x rs j ( t + 1) = x rs � y rs � y rs � y rs jk ( t ) ≤ x rs j ( t ) + ij ( t ) − jk ( t ) j ( t ) � � � i ∈A k ∈A k ∈ Γ + � j Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 9

Vehicle queueing model x rs j ( t ) is the number of vehicles on link j traveling from r to s p r ( t ) is the number of vehicles parked at r � � x rs � j ( t ) + p r ( t ) = F j ∈A ( r,z ) ∈Z 2 r ∈Z Conservation of link queues: � � � x rs j ( t + 1) = x rs � y rs � y rs � y rs jk ( t ) ≤ x rs j ( t ) + ij ( t ) − jk ( t ) j ( t ) � � � i ∈A k ∈A k ∈ Γ + � j Conservation of parked queues: � � y qr � � y rs p r ( t + 1) = p r ( t ) + ir ( t ) − rj ( t ) i ∈A q ∈Z j ∈A s ∈Z Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 9

Markov decision process Queues of passengers and vehicles define a Markov chain.     w rs ( t + 1) = w rs ( t ) + d rs ( t ) − min � y rs rj ( t ) , w rs ( t )  j ∈A  y qr � � � � y rs p r ( t + 1) = p r ( t ) + ir ( t ) − rj ( t ) i ∈A q ∈Z j ∈A s ∈Z � � x rs j ( t + 1) = x rs y rs y rs j ( t ) + ij ( t ) − jk ( t ) i ∈A k ∈A Since vehicle movements can be controlled, this is a Markov decision process model. Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 10

Stability and passenger service ( r,s ) ∈Z 2 w rs ( t ) is the number of waiting passengers at time t � ( r,s ) ∈Z 2 w rs ( t ) will increase over time If demand is unserved, then � Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 11

Stability and passenger service ( r,s ) ∈Z 2 w rs ( t ) is the number of waiting passengers at time t � ( r,s ) ∈Z 2 w rs ( t ) will increase over time If demand is unserved, then � Definition The stochastic queueing model is stable if there exists some K < ∞ s.t. T 1 � � E [ w rs ( t )] ≤ K ∀ T ∈ N T t =1 ( r,s ) ∈Z 2 Equivalently, ∃ Lyapunov function ν ( w ( t )) ≥ 0 s.t. E [ ν ( w ( t + 1)) − ν ( w ( t )) | w ( t )] ≤ κ − ǫ | w ( t ) | for all w ( t ) for κ < ∞ , ǫ > 0 . Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 11

Assumptions SAV travelers wait in the system until served ◮ If SAV travelers exited, the concept of stability would need to be redefined. Constant travel times for vehicles Entire SAV fleet can be centrally dispatched Queueing model Max-stability dispatch for SAVs M. W. Levin, D. Kang 12

Maximum-stability policy

Max-pressure policy with maximum stability T 1 � � w rs ( t ) f rs ( t + τ ) max T τ =1 ( r,s ) ∈Z 2 � f rs ( t + τ ) ≤ p r ( t + τ ) s . t . s ∈Z f qr � t + τ − Φ r � p r ( t + τ + 1) = p r ( t + τ ) + � − q q ∈Z x qr � f rs ( t + τ ) + � � i ( t + τ − Φ r i ) s ∈Z q ∈Z i ∈A f rs ( t + τ ) ≥ 0 T is the planning horizon — how far we look ahead f rs ( t + τ ) anticipates future vehicle dispatch p r ( t + τ ) anticipates future vehicle availability Maximum-stability policy Max-stability dispatch for SAVs M. W. Levin, D. Kang 14

Planning horizon analysis T 1 � � w rs ( t ) f rs ( t + τ ) max T τ =1 ( r,s ) ∈Z 2 � f rs ( t + τ ) ≤ p r ( t + τ ) s . t . s ∈Z f qr � t + τ − Φ r � p r ( t + τ + 1) = p r ( t + τ ) + � − q q ∈Z x qr � f rs ( t + τ ) + � � i ( t + τ − Φ r i ) s ∈Z q ∈Z i ∈A f rs ( t + τ ) ≥ 0 T is the planning horizon — how far we look ahead T must be large enough to dispatch vehicles across the network. At least � � Φ r max . q r Maximum-stability policy Max-stability dispatch for SAVs M. W. Levin, D. Kang 15

Stability region What demand rates ¯ d ∈ D could be served by any SAV dispatch policy? We want to serve any ¯ d ∈ D with the max-pressure policy. Stability proof Max-stability dispatch for SAVs M. W. Levin, D. Kang 16

Stability region What demand rates ¯ d ∈ D could be served by any SAV dispatch policy? We want to serve any ¯ d ∈ D with the max-pressure policy. Average SAV flow rates from r to s are enough to serve average demand: ri ≥ ¯ ∀ ( r, s ) ∈ Z 2 � y rs d rs ¯ i ∈ Γ + r Stability proof Max-stability dispatch for SAVs M. W. Levin, D. Kang 16

Stability region What demand rates ¯ d ∈ D could be served by any SAV dispatch policy? We want to serve any ¯ d ∈ D with the max-pressure policy. Average SAV flow rates from r to s are enough to serve average demand: ri ≥ ¯ ∀ ( r, s ) ∈ Z 2 � y rs d rs ¯ i ∈ Γ + r Constraints on average SAV flow rates: � � y qr � � y rs ¯ ir = ¯ ∀ q ∈ Z jr q ∈Z s ∈Z j ∈ Γ + i ∈ Γ − r r � � y rs y rs ∀ ( r, s ) ∈ Z 2 , ∀ j ∈ A o ¯ ij = ¯ jk j ∈ Γ + i ∈ Γ − j j � � y rs ¯ ij ≤ F ( r,s ) ∈Z 2 ( i,j ) ∈A 2 Stability proof Max-stability dispatch for SAVs M. W. Levin, D. Kang 16