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it was recently reported that over one year in a small los angeles business district , cars cruising for parking burned 47,000 gallons of gasoline and produced 730 tons of carbon dioxide @xcite . meanwhile , the consulting firm mckinsey recently claimed that the average car owner in paris spends four years of his or her life searching for a parking space @xcite . the parking assignment problem associated with electric vehicles becomes even more acute . due to the limited range of these vehicles , the marginal cost of expending energy to search for spaces may , in some cities , be prohibitively high . thus , there is a real and compelling societal and economic need to revisit parking . increases in car ownership , inadequate public and private parking facilities , and synchronised demand , have led to serious temporal mismatches in parking space supply and demand . finding a parking space at certain times of the day is a non - trivial challenge . furthermore , finding spaces is not only costly in terms of wasted time for the driver , but it also creates congestion and pollution affecting everyone . thus , improving parking improves both economic efficiency and environmental quality . indeed , already major companies and cities are responding to these challenges . for example , many city authorities , in order to safeguard resident on - street parking from people commuting to a city through the day , have initiated schemes such as resident permit parking , and a number of commercial initiatives have emerged in the parking area . sfpark and justpark provide examples of city authorities and companies investing heavily in parking research and products within a smart cities context .    in this work , our objective is not to revisit prior work on the topic . rather , we propose a new solution that is widely applicable in cities . we consider two nearby entities that have complementary supply and demand , i.e. , there is a shortfall of parking space at one and a surplus at the other . _ we call the entity with a shortfall of parking a _ mini - city_. many examples come to mind : university campuses , technology parks , shopping centers , and government facilities located in the suburbs . in our work , we use the university campus as an example . the second entity is the area surrounding the mini - city . during business hours , while parking spaces are limited in the mini - city , many nearby residents leave home for work . thus , there is an opportunity for the mini - city to use these vacated parking spaces , as well as from schools , hotels , apartment complexes and so on . consequently , the mini - city has access to two classes of parking spaces : _ premium _ spaces , e.g. , those located on the university campus ; and _ secondary _ spaces located nearby and perhaps serviced by a shuttle . note that the secondary spaces are not owned by the mini - city but rather are rented from secondary parties or _ landlords_. given this basic scheme , we consider two specific design issues . first , we wish to guarantee a quality of service for the landlords by setting aside reserve spaces in the mini - city as contingency for events where secondary spaces are suddenly unavailable . second , we wish to ensure that the premium spaces are allocated optimally among _ users _ ( drivers ) while preserving each user s privacy . within the research community , questions concerning how to manage parking space supply - demand mismatch are actively being investigated from a variety of different angles . one important aspect concerns the delivery of up - to - date , accurate , real - time information to parking systems in order to achieve greatest system efficiency . a car park monitoring and vacancy detection system is discussed in @xcite and consists of a camera and image analysis system that reports statistics . the aim for such systems is to provide timely information on available parking spaces and minimize search time . parkya is an early - stage company offering a service that signposts parking locations and allows users to pay for their parking through a smartphone application . driving directions to the parking spot are additionally available . in the near future , such services should benefit greatly from incorporating increased amounts of real - time data , concerning up - to - date details on current parking vacancies , and predicted travel times given current road conditions . the system of @xcite monitors and reports the state of availability of parking spaces and additionally uses context information generated by the city and its citizens to provide accurate responses to demand . a predictive approach is proposed in @xcite and further studied in @xcite . the authors develop a method in which car parks are able to count the number of arriving and departing cars , and report these to participating cars . this allows them to predict the likelihood of a parking space being available at the estimated time that the car will arrive there . this work uses ideas from queueing theory to predict the occupancy upon arrival , with car parks being modelled as single server queues with a poisson arrival process and exponentially distributed service times . it should be mentioned that this significant reduction in requirements by using a stochastic approach comes at the cost of certainty for the customers . the lack of a reservation system makes it possible that customers arrive to a fully occupied car park . the main drawback of the approach is , however , that ultimately the customers will want to use the information to make a decision whether to try their luck and drive to the car park or to go somewhere else . accordingly , there is feedback embedded in the system which needs to be taken into consideration ; namely , when users choose to drive to a car park based on the predictions made , they then affect the arrival process , rendering the model and predictions no longer valid . this latter issue is addressed in @xcite , where multiple car park load balancing is achieved using randomisation based on congestion signals from the participating car parks . significant extensions of this work are given in @xcite . certainly , access to real - time information , on its own , is not enough . given the delay between an user identifying and driving to a parking space , and the fact that other users may also compete for the same parking space , a successful parking system would benefit from either intelligent assignment or reservation capability to reduce localized congestion @xcite . the reservation system is championed in @xcite , where an architecture for an intelligent parking assistant is proposed as a public car park management solution . in @xcite , the parking problem is viewed as a dynamic resource - allocation problem . similarities to problems in communication networks are drawn , for which a host of tools and methods have been developed over the last decades . an online reservation system is proposed , where cars communicate their parking requirements and are assigned a parking space , which is then reserved and can not be used by any other vehicle . a similar approach is proposed in @xcite , albeit with a different assignment routine , and also allows the user to specify the price that he is willing to pay . the main focus of the paper is revenue maximisation , but it is also possible to achieve other goals , such as reducing traffic levels or ensuring some sort of fairness between users from different social classes .    in @xcite , an agent - based simulation model for parking facilities management was developed , the goal of the tool being to provide better understanding of how complex urban traffic systems evolve under different parking pricing strategies and different levels of parking enforcement . agents included drivers , parking authorities , law enforcement and city government . models for cruising - for - parking behaviour were developed in @xcite . in @xcite , the quality of service ( qos ) metrics is used to dimension a shared vehicle fleet such that satisfactory levels of access to the fleet were ensured to its pool of users . while the topic of @xcite was not parking per se , the paper inspires some of the notions of our current work ; that is , we consider the dimensioning of a special subset of parking spaces in our mini - city that we will call the _ reserve_. we discuss this problem in the following sections of the paper . we will consider the following scenario . imagine a university campus with a total of @xmath0 parking spaces , surrounded by a total of @xmath1 private parking spaces ( e.g. , from residential complexes ) , as illustrated in figure [ problem_setup ] . a typical working day sees the university parking fill to capacity with vehicles belonging to students and staff , to the extent that the @xmath0 spaces can not meet the demand . some campus arrivals thus have to search elsewhere . at the same time , many nearby residents drive to work during the day and vacate their @xmath1 private parking spaces . what we have is a wasted resource in one area ( i.e. , residential parking spaces ) and a stressed resource nearby ( i.e. , the on - campus parking ) . we assume that a contract exists between the university and the _ landlords _ or owners of these @xmath1 private parking spaces . the contract stipulates that these landlords lease their driveways to the university during daytime in the sequel , we will denote the time period in which the extra spaces are leased by @xmath2 $ ] . this can be possible because the landlords drive to work during this time period .    in our nomenclature , the premium resources are the on - campus parking spaces , while the secondary resources are the contracted residential parking spaces . we will now consider how to design a parking system , bearing in mind the needs of the mini - city community , and the needs of the landlords . we shall consider the following facets of the design problem :    1 . how do we accommodate the needs of parking space owners ( landlords ) given that situations may arise when landlords will need to return home during the contracted interval ? an example is scheduling a repairman to visit the home to fix a fitting . ( see section [ maths ] . ) 2 . how do we accommodate the needs of the landlords given that situations arise when misbehaving university members will not vacate the parking space outside of the contracted interval ? ( see section [ maths ] . ) how do we allocate efficiently access to the premium spaces to the university community in a manner that preserves the privacy of individuals in the community ? ( see section [ fair ] . )    as we shall see , we answer these questions in a stochastic framework by developing suitable quality of service ( qos ) metrics to dimension the aforementioned parking system . to address items 1 and 2 , we shall set aside a subset of @xmath3 premium spaces as a reserve . the research question in this latter context then becomes how to use the qos metrics to dimension the size of @xmath3 . we shall address item 3 by applying recent ideas from distributed resource allocation . * comment : ( generalization ) * the parking scenario presented in this paper is very specific , but our method can also be applied in the following general setting . there are two resources , @xmath4 and @xmath5 . these resources are atomic ( indivisible ) , with each atom being allocated to one user . each user has a preference over the two resources . using resources @xmath4 and @xmath5 decreases a certain qos metric . in the following main sections , we answer two questions :    * given a qos requirement , how much of resource @xmath4 and resource @xmath5 should we use ? * given an efficiency metric derived from the user preferences , how do we allocate the resources optimally among users in a setting where the allocation is repeated over multiple iterations ? in this section , we consider the problem of dimensioning @xmath6 premium spaces as _ reserve _ in order to provide sufficient quality of service ( qos ) guarantees to the landlords of the secondary spaces . let @xmath7 index the landlords of the parking spaces . for each such landlord @xmath8 , we define a non - negative valued random variable @xmath9 , which denotes the time at which he or she returns home and needs to get the parking space back . under normal circumstances , @xmath9 is greater than @xmath10 , but on rare occasions a landlord may choose to come home early .    for simplicity , we assume that each parking space @xmath8 has exactly one daytime user each day . for each parking space @xmath8 , we also define a non - negative valued random variable @xmath11 , which denotes the departure time of the daytime user of the space @xmath12 . under normal circumstances , @xmath11 is less than @xmath10 , but we assume a small number of miscreants so that not all spaces are always vacated on time .    for convenience , recall that @xmath2 $ ] denotes the nominal rental window for every parking space . in other words , the landlord of space @xmath12 agrees to park only outside the interval @xmath2 $ ] , whereas the daytime user of space @xmath12 agrees to park during the interval @xmath2 $ ] only . we define the following two events for each secondary parking space @xmath8 : @xmath13 \ } \cap \{t_i < a_i\},\\    o_i \triangleq \{w < t_i\ } \cap \{t_i < a_i\ } .\end{aligned}\ ] ] each of these events represents an outcome where a landlord would like to use the space @xmath12 , but can not do so . the _ home - early _ event @xmath14 is due to the landlord needing the space during the day . the _ overstay _ event @xmath15 is due to the fact that the daytime user overstayed .    for easy of presentation , we assume the following properties concerning the above random variables . [ as:1 ] the random variables @xmath16 are independent and identically distributed . likewise , @xmath17 are also i.i.d . moreover , all these random variables are mutually independent . finally , we assume that all the distributions have densities . we begin by quantifying the probability of an event @xmath15 in terms of the probability of daytime users overstaying . recall that the distribution of the random variable @xmath11 characterises the probability of the daytime user overstaying in space @xmath12 . first , observe that @xmath18 next , we derive an exact expression for @xmath19 using the independence assumption ( assumption  [ as:1 ] ) . let @xmath20 denote the probability distribution of @xmath21 ; and @xmath22 denote the probability distribution of @xmath23 . under assumption  [ as:1 ] , we have @xmath24    we estimate @xmath20 from data in the following section observe that @xmath25 the claim follows by definition of the distribution @xmath26 of @xmath23 . now we derive a formula that we can use to dimension the reserve parking space @xmath3 from the contingent of @xmath0 premium spaces . recall that there are @xmath3 reserve parking spaces ( the reserve `` buffer '' ) set aside by the university . in this section , we consider the probability @xmath27 of the event that more than @xmath3 spaces are needed to accommodate landlords needing the reserve buffer during daytime @xmath2 $ ] : @xmath28 where @xmath29 denotes a bernoulli random variable taking value 1 when event @xmath14 occurs , and value 0 otherwise . first , we characterise the probability of the event @xmath30 in terms of the probability distributions of @xmath23 and @xmath21 . let @xmath22 denote the probability distribution of @xmath23 . let @xmath20 denote the probability distribution of @xmath21 . under assumption  [ as:1 ] , we have @xmath31    @xmath22 and @xmath20 are estimated from data in the next section . let @xmath32 . observe that @xmath33 \ } \right)\\         \mbox{(by non - negativity of $ t_1$)}\quad&= { { \mathbb p}}\left ( \{t_1 < a_1\ } \cap \ { t_1 < w \ } \right)\\         & = { { \mathbb p}}\left ( \{t_1 < a_1\ } \cap \ { t_1 < w \ } \mid a_1 \leq           w\right ) { { \mathbb p}}(a_1 \leq w)\\ & \quad+ { { \mathbb p}}\left ( \{t_1 < a_1\ } \cap \ { t_1 < w \ } \mid a_1 >           w\right ) { { \mathbb p}}(a_1 > w)\\         \mbox{(by bayes ' rule)}\quad&= { { \mathbb p}}\left ( t_1 < a_1 \mid a_1 \leq           w\right ) { { \mathbb p}}(a_1 \leq w ) + { { \mathbb p}}\left ( t_1 < w \right ) { { \mathbb p}}(a_1 > w)\\         & = { { \mathbb p}}\left ( t_1 < a_1 \leq           w\right ) + { { \mathbb p}}\left ( t_1 < w \right ) { { \mathbb p}}(a_1 > w)\\    & = \int_{t=0}^w { { \mathbb p}}\left ( t < a_1 \leq           w\right ) df_{t}(t ) + { { \mathbb p}}\left ( t_1 < w \right ) { { \mathbb p}}(a_1 > w)\\    & = \int_{t=0}^w ( f_{a}(w ) - f_{a}(t ) ) df_{t}(t ) + f_{t}(w ) ( 1-f_{a}(w)),\end{aligned}\ ] ] which is the claim .    as a corollary , we have the following expression for the probability that setting @xmath3 reserve spaces at the university is not enough . let @xmath32 . under assumption  [ as:1 ] , @xmath27 is a random variable entirely characterized by @xmath34 : @xmath35[cor_5 ]    observe that this probability can be mitigated by increasing the parameter @xmath3 . given a qos target @xmath27 equation can be used to determine the corresponding value of @xmath3 needed to achieve it . this is illustrated through an example in the next subsection .      in the previous section we have considered arbitrary probability distributions @xmath26 , @xmath36 in the formulae derived . in this section , we give estimates @xmath37 of these distributions using publicly available data . given samples @xmath38 from the distribution @xmath26 , the corresponding empirical distribution - estimate takes the form @xmath39}.\end{aligned}\ ] ]    first , we estimate the distribution @xmath36 . recall that @xmath11 is the random variable denoting the duration of use of the @xmath12th secondary parking space . to derive an estimate , we use data on parking space utilisation collected in the city of dublin . each data point corresponds to the time duration of one parking event . the histogram distribution of these durations is shown in figure [ data ] . of course , we can obtain a better estimate of the distribution of parking usage in a university campus if we have access to more particular data . based on figure  [ data ] , in order to simulate the fact that 5% of users of secondary parking spaces overstay , we set @xmath40 for our example . next , we estimate the distribution @xmath26 for @xmath41 . this distribution accounts for landlords who do not leave home and who arrive home normally after working hours . there are many reasons for landlords to return home early , or not leave home at all . for simplicity , we estimate the frequency of such days with the number of sick days of nhs staff in england over the period from april 2009 to february 2014 . this data is presented in figure [ data2 ] . for simplicity , we use the average sickness absence rate to estimate the probability of @xmath42 , and assume that @xmath43 otherwise ( cf . table  [ fti ] ) .        .probability distribution function of @xmath9 . for simplicity , we assume that the function is constructed so that 4.2% of landlords remain at home due to sickness , while the other 95.8% return home exactly at time instant @xmath10 . [ cols="^,^",options="header " , ]     [ fti ]    finally , we use corollary [ cor_5 ] to perform a dimensioning exercise based on our data . figure [ dimensioning_example ] illustrates the probability @xmath27 that @xmath3 reserve spaces are insufficient when @xmath1 secondary spaces are contracted . the probability @xmath27 eventually falls exponentially fast versus @xmath3 . the dependence of @xmath27 on @xmath1 is more subdued . in other words , for a fixed value of @xmath27 , a linear increase in the number of secondary spaces only requires a logarithmic increase in the number of reserve spaces .     that @xmath3 reserve spaces are insufficient when @xmath1 secondary spaces are contracted ] in this section , we consider the point of view of the user ( i.e. a staff member or student ) who requires parking at the university . typically , such a user would purchase a monthly or yearly parking ticket . this ticket then provides them with the opportunity to compete for a university parking space . however , such a ticket does not necessarily guarantee them a parking space on the university grounds . that is , if they arrive to the university `` too late '' on any given day , the car parks on - campus may already be full . this is because more parking tickets are sold than there are actual spaces to park . such `` first come , first served '' systems can be inefficient . consider the example of a studying mother or father who , even though they paid the same amount for a monthly or yearly parking ticket as everyone else , must first always take their children to school in the morning . they thus always arrive to the campus a little bit later than everyone else and have a greater chance of missing out on an on - campus parking space . we are interested in a scheme that offers equality in regards to access to the premium , on - campus parking spaces for all university users over the long - term period of validity of their parking ticket .    for our scheme , we suppose that the university has been able to obtain enough apartment building parking spaces such that @xmath44 is greater than or equal to the number of university users requiring a parking space on any given day . the problem then is how to allocate the premium and secondary parking spaces to users efficiently over time . how do we ensure that one campus user does not have to park off - campus for the majority of the time while another user is almost always offered a car space on the university grounds ? our solution to this problem is motivated by additive - increase / multiplicative - decrease ( aimd ) optimisation algorithms @xcite . we now consider the problem of providing efficient access to the available premium spaces at the mini - city . these spaces refer to the @xmath45 spaces available after allocating @xmath3 of the spaces as reserve according to the previous section . for simplicity of exposition , we shall assume that @xmath46 in this section , but the results generalize in a straightforward fashion . we introduce and recall the following notation to describe the problem data and the variables used in our algorithm .    * an integer denoting the number of available premium parking spaces . * an integer denoting the number of users wishing to avail of premium parking spaces . we assume that @xmath47 . * denotes discrete time , @xmath48 . in our interpretation this corresponds to the number of days the system is operating . for convenience , we assume spaces are assigned on a per - day basis , but the general principles of the algorithm do not depend on this assumption . * this is a state variable associated with the @xmath12th user . it takes the value @xmath49 if this user is given access to a premium parking on the @xmath50th day and zero otherwise . * this is a average access for the @xmath12th user up to the @xmath50th day : @xmath51    it is possible to formulate the premium parking space allocation problem in several ways . for example , one could require that that the long - term average admission to the premium parking space is equal for all users , i.e. for all @xmath52 @xmath53 this assumes that all users are equal in the desire to access the premium parking space . here we follow a more general approach and assume that each user @xmath12 has a cost function @xmath54 \to \mathbb r$ ] . for a frequency @xmath55 $ ] of premium space allocation , the cost @xmath56 represents the monetary inconvenience cost that user @xmath12 experiences from @xmath57 . this function specifies the priority that this user is assigned . it could represent the amount a user is willing to pay , or it could be related to the number of passengers carried by this users , or the access that this user has to public transportation ( meaning that users with fewer possibilities for alternative transport should have prioritised access to parking spaces ) . given these individual cost functions , our aim is to design a system that achieves overall minimal cost for the group . we formulate the optimal allocation of resources as a minimization problem : @xmath58    our proposed simple algorithm for solving the parking allocation problem can be summarized as follows . we assume that every event each user is allocated access to the premium spaces by tossing a coin . for example , one embodiment of this idea is to use a smart - phone application . more specifically , each user is assigned a cost function by a government authority . for example , this could be based on vehicle class , disability , need for childcare , etc . on every subsequent day , the application outputs a variable that determines whether the user is assigned a premium space . access to the space is assigned to the @xmath12th user with the following probability : @xmath59 a central authority that owns the parking spaces , calculates @xmath60 based on past utilization and broadcasts this scalar to all participating vehicles . the following example illustrates the performance of this simple algorithm . we will assume that the functions @xmath61 are continuously differentiable and strictly convex ; so that in particular the optimal point @xmath62 satisfying the constraints is unique . furthermore , we introduce an assumption which ensures that the optimal point @xmath63 has only positive entries . this assumption also guarantees that the algorithm we will describe is well defined for every user . we wish to control the access to the premium space in such a way that the average utilization for each user approaches the optimal value @xmath63 , i.e. for large @xmath50 we want to achieve @xmath64 subject to the ( loose ) capacity constraint @xmath65 . that is , all premium spaces are occupied , on average . further , we wish to do this in a manner that preserves the privacy of users . that is , we do not wish to reveal @xmath66 and @xmath61 to any other users in the course of the optimization . finally , the necessary communication between the users should be minimal so as not to create a communication overhead that would be hard to sustain in an uncertain environment where users can not be expected to participate at all times .    in what follows the mechanism for preserving privacy is to develop a distributed algorithm . we loosely follow the ideas in @xcite , where a distributed stochastic algorithm is presented which guarantees that the average utilization variables @xmath67 converge to the optimal points @xmath68 , which is also called almost sure convergence in a stochastic context . ] . the algorithm presented here extends the ideas of @xcite , however , because we wish to ensure in addition that the instantaneous utilization variables @xmath69 sum to @xmath0 , or at least to a value close to it . moreover , the resource to be allocated in our setting is atomic , as opposed to arbitrarily divisible . these differences require substantial changes to the algorithm presented in @xcite .    at each time @xmath50 , each user @xmath12 determines a probability @xmath70 and sets @xmath71 where we note that all users make this probabilistic choice independently of other users or previous decisions . the evolution of the probabilities is governed by the equation @xmath72 note that each user @xmath12 can determine its own probability with the exclusive knowledge of its own past utilization @xmath73 and cost function @xmath61 . no information from other users is required . the scalar @xmath60 is a network wide constant determined by the central agency and broadcast to all users . here , @xmath60 is chosen such that @xmath74 for all @xmath75 and all @xmath76 . it is determined in a time - varying manner as it also influences the demand for premium spaces . specifically , if at a certain time @xmath50 each @xmath70 is fixed then the expected utilization of the premium spaces is @xmath77 moreover , the ( random ) instantaneous utilization @xmath78 is concentrated around the expected utilization by independence and hoeffding s inequality . if we wish to ensure optimal utilization of the premium spaces and avoid overbooking , the expected utilization should be below the number of premium parking spaces . for instance , a standard deviation below this number , depending on the desired quality - of - service metric . denoting this number by @xmath79 , we will therefore adjust @xmath60 so that the expectation in tracks @xmath80 . as the expectation is unknown , we use the observed utilization as an estimator for this . taking a simple error regulation approach we thus arrive at @xmath81    the overall system is now prescribed by the dynamics of @xmath82 given by , the dynamics of @xmath83 given by and the dynamics of @xmath84 as described in . a proof of the convergence of the algorithm is beyond the scope of this paper . however , we present below an example to show its efficacy .       for three users.,height=240 ]            , height=240 ]        we simulate a population of 900 users competing for 450 premium parking spaces . each evening , users are assigned a parking space as described above ( with the scalar @xmath60 determined using a pi control ) . for simplicity , users have one of three cost functions : @xmath85 , @xmath86 , and @xmath87 , all of which are strictly convex . figure [ fig : ap ] shows the average utilisation achieved for each class of vehicle . figure  [ fig : pu ] and  [ fig : occ ] shows that the average utilisation of premium spaces is concentrated around the target utilisation of 450 . figure [ fig : gain ] shows the evolution of the controller gain and figure [ fig : con ] shows that the cost functions derivatives do actually achieve consensus asymptotically .      here we will briefly comment on the heuristic which suggests that the algorithm works in a general setting . to this end we have to point out that a number of assumptions are required . first , in our derivation of the update rule we will rely on an infinitesimal characterization of the optimal point . this requires that the optimal point does not lie on the boundary of the constraint set . thus in the following we will assume that the optimal point @xmath63 satisfies @xmath88    second , the gain parameter @xmath89 in the feedback scheme needs to be chosen in such a manner that the system is stable . noting that by the expectation @xmath90 is a multiple of @xmath60 , we choose @xmath91 where the maximum with respect to @xmath92 is taken over the constraint set . we then reformulate to obtain @xmath93 in the previous formulation , the bracket giving the factor of @xmath60 needs to be in the interval @xmath94 sufficiently often to ensure stable behaviour of the system and thus the desired tracking property . the choice of @xmath89 in is made to ensure that the desired stability property holds . this ensures convergence of the sequence @xmath60 .    to argue for the convergence of @xmath95 we follow the heuristic presented in @xcite . the optimisation problem is a constrained problem for which we have assumed that the boundary conditions @xmath96 are not active . it is then classical to introduce the lagrange parameter @xmath97 and consider the lagrangian @xmath98 from the karush - kuhn - tucker ( kkt ) conditions ( * ? ? ? * section 5.5.3 ) , the following necessary and sufficient condition for optimality can be obtained by setting all partial derivatives to zero . in the optimal point @xmath99 , @xmath100 we have @xmath101 in other words , the system is at optimality when the derivatives of the cost functions are in consensus . this observation is at the core of the choice of the probability functions in as we now explain .    assuming that the system described by , , and has an ergodicity property we obtain the long - term average for the access to the premium car park satisfies ( in the limit ) @xmath102 where @xmath103 is the steady state probability that user @xmath12 has access to the resource . given this assumption , we claim that our choice of probability functions @xmath70 is such that the equation for the steady state behaviour of our system is equivalent to the kkt condition . by each user has a probability of access that satisfies @xmath104 where the constant of proportionality is the same for all users . suppose now that after a transient period we have @xmath105 , the steady state long - run average which exists under our ergodicity assumption . then we can write @xmath106 . thus , in steady state , @xmath107 and so @xmath108 for all @xmath109 . so in the limit we expect @xmath110 for all @xmath12 . this means that in the limit we converge to a situation where the derivatives of the cost functions are in consensus . as we have seen from the kkt conditions the consensus conditions characterizes the optimal point of the optimization problem so that we obtain @xmath111 thus the average utilization converges to the optimal point . several comments are merited . * this is a private algorithm . users do not communicate with other users , nor do they reveal any state information or cost information to the central authority . * this is a randomized algorithm . thus we only have @xmath112 . we argue that small fluctuations in the occupancy level about the capacity is acceptable since some users may not use the premium spaces , even if allocated one . furthermore , in the event of all users availing of their allocated premium spaces , extra space is always available in the secondary spaces , and in the reserved spacing . that is , the instantaneous allocation @xmath113 may exceed the capacity @xmath0 for some time steps @xmath50 , in which case , some users will have to be re - routed to the secondary parking spaces . * the algorithm can be modified such that the parameter @xmath84 does not update at every iteration , but once in a while ( i.e. , from day to day ) . * the proposed allocation scheme can be applied to more general problems of the form of . for example , the same problem arises also in assigning space in overhead bins on passenger planes , seats in trains , etc . in all of these examples , users pay a fee to compete for access to a limited resource . however , access to the resource is not guaranteed and in effect the resource is allocated in a first - come , first - served manner . from a societal viewpoint , customers with constraints , e.g. dropping children at school , are almost always disadvantaged by such schemes . in this work , we have modeled two aspects of parking system design and proposed two corresponding solutions . first , we propose to complement primary or premium parking spaces with a large number of secondary parking spaces . these spaces are obtained through contracts for shared use that provide probabilistic qos guarantees at the expense of a relatively small number of reserve premium spaces . secondly , we propose a distributed algorithm for repeated allocation of premium spaces , which guarantees that , over time , the users who benefit most from such spaces obtain them proportionally more frequently . the main feature of this algorithm is to keep private all information related to the benefit derived by individual users . the next step in our research agenda is to deploy the described system on a university campus . from an algorithmic perspective , it remains an open problem to quantify the rate of convergence of @xmath114 and to situations where we have dynamic arrivals and departures throughout each day . another open problem is to quantify the probability of overshooting the capacity : i.e. , the event @xmath115 . it is also useful to consider a version of the allocation algorithm for the case where a different subset of users competes in each day . w. elfrink , the smart - city solution . article , published by mckinsey , october 2012 . available at : http://www.mckinsey.com/insights/public_sector/the_smart-city_solution . 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parking spaces are resources that can be pooled together and shared , especially when there are complementary day - time and night - time users . 
 we answer two design questions . 
 first , given a quality of service requirement , how many spaces should be set aside as contingency during day - time for night - time users ? 
 next , how can we replace the first - come - first - served access method by one that aims at optimal efficiency while keeping user preferences private ?