Allocation in a Vertical Rotary Car Park

Abstract

We consider a vertical rotary car park consisting of l levels with c parking spaces per level. Cars arrive at the car park according to a Poisson process, and if there are parking spaces available, they are parked according to some allocation rule. If the car park is full, arrivals are lost. Cars remain in the car park for an exponentially distributed length of time, after which they leave. We develop an allocation algorithm that specifies where to allocate a newly-arrived car that minimises the expected cumulative imbalance of the car park. We do this by modelling the working of the car park as a Markov decision process, and deriving an optimal allocation policy. We simulate the operation of some car parks when the policy decision making protocol is used, and compare the results with those observed when a heuristic allocation algorithm is used.

Appendix

1.1 Notation

$$ \displaystyle\begin{array}{rcl} \begin{array}{ll} l\ &\qquad \mbox{ Number of levels in the car park} \\ c\ &\qquad \mbox{ Number of cars per level} \\ \phi &\qquad \mbox{ A configuration of the car park} \\ \#\phi (L)\ &\qquad \mbox{ Numbers of cars on the left hand side of the car park} \\ &\qquad \mbox{ when in configuration $\phi $} \\ \#\phi (R)\ &\qquad \mbox{ Numbers of cars on the right hand side of the car park } \\ &\qquad \mbox{ when in configuration $\phi $} \\ \nu \ &\qquad \mbox{ Number of turns}\\ \varTheta \ &\qquad \mbox{Imbalance measure} \\ \tau \ &\qquad \mbox{ Turning penalty} \\ I\ &\qquad \mbox{ Imbalance} \\ X(t)\ &\qquad \mbox{ A homogeneous, finite-state, discrete-time stochastic process} \\ S\ &\qquad \mbox{ A state space} \\ m\ &\qquad \mbox{ Number of states in the state space} \\ i,j\ &\qquad \mbox{ States in the state space} \\ A(i)\ &\qquad \mbox{ Set of actions that can be taken in state $i$} \\ a\ &\qquad \mbox{ An action} \\ p(j\vert i,a)\ &\qquad \mbox{ Probability of moving from state $i$ to state $j$ if action $a$ is taken} \\ r(j\vert i,a)\ &\qquad \mbox{ Reward received in moving from state $i$ to state $j$ if action $a$ is taken} \\ r(i,a)\ &\qquad \mbox{ Expected reward received in moving from state $i$ if action $a$ is taken} \\ V _{t}(i)\ &\qquad \mbox{ Maximum expected reward that can be accrued from time $t$ until } \\ &\qquad \mbox{ time step $T$, given that the state of the system at time step $t$ is $i$} \\ \lambda \ &\qquad \mbox{ Cars' Poisson arrival rate}\\ \mu \ &\qquad \mbox{Parameter for the exponentially distributed car parking time} \\ t\ &\qquad \mbox{ The number of events (arrivals and departures)} \\ T\ &\qquad \mbox{ The event number at which the process ceases} \\ \phi _{t}\ &\qquad \mbox{ State of the car park after $t$ events} \\ \vert \phi _{t}\vert \ &\qquad \mbox{ Number of cars in the car park after $t$ events} \\ \rho \ &\qquad \mbox{ Load on the car park} \\ A(\phi _{t})\ &\qquad \mbox{ Set of all states accessible from state $\phi _{t}$ if an arrival occurs} \\ d_{s}(\phi _{t})\ &\qquad \mbox{ State that is moved to if a car departs from space $s$ when the car park } \\ &\qquad \mbox{ is in state $\phi _{t}$} \\ V _{t}(\phi _{t})\ &\qquad \mbox{ Minimum expected cumulative imbalance from event $T - t$ to $T$ if the } \\ &\qquad \mbox{ car park is in state $\phi _{t}$} \\ \epsilon \ &\qquad \mbox{ Predetermined tolerance level}\end{array} & & {}\\ \end{array} $$

1.2 MDP Convergence Results

See Table 12.2.

1.3 Policy When l = 8 and c = 1

See Tables 12.3 and 12.4.

1.4 Simulation Results for l = 8 and c = 3

See Tables 12.5 and 12.6.

1.5 Simulation Results for l = 20 and c = 1

See Tables 12.7 and 12.8.