Optimal Vehicle Dimensioning for Multi-Class Autonomous Electric Mobility On-Demand Systems

Optimal V ehicle Dimensioning for Multi-Class Autonomous Electric Mobility On-Demand Systems Syrine Belakaria ∗ , Mustafa Ammous ∗ , Sameh Sorour ∗ and Ahmed Abdel-Rahim † ‡ ∗ Department of Electrical and Computer Engineering, Univ ersity of Idaho, Moscow , ID, USA † Department of Civil and En vironmental Engineering, University of Idaho, Mosco w , ID, USA ‡ National Institute for Advanced T ransportation T echnologies, Univ ersity of Idaho, Moscow , ID, USA Email: { ammo1375, bela7898 } @vandals.uidaho.edu, { samehsorour , ahmed } @uidaho.edu Abstract —A utonomous electric mobility on demand (AEMoD) has recently emerged as a cyber -physical system aiming to bring automation, electrification, and on-demand ser vices f or the futur e private transportation market. The expected massive demand on such services and its resulting insufficient charging time/resources prohibit the use of centralized management and full vehicle charging. A f og-based multi-class solution for these challenges was recently suggested, by enabling per-zone management and partial charging for differ ent classes of AEMoD vehicles. This paper focuses on finding the optimal vehicle dimensioning f or each zone of these systems in order to guarantee a bounded r esponse time of its v ehicles. Using a queuing model repr esenting the multi-class charging and dispatching processes, we first derive the stability conditions and the number of system classes to guarantee the response time bound. Decisions on the pr oportions of each class vehicles to partially/fully charge, or directly serve customers ar e then optimized so as to minimize the vehicles in-flow to any given zone. Excess waiting times of customers in rare critical events, such as limited char ging resources and/or limited vehicles av ailabilities, are also in vestigated. Results show the merits of our proposed model compared to other schemes and in usual and critical scenarios. Keyw ords — Autonomous Mobility On-Demand; Electric V ehi- cles; F og-based Architecture; Dimensioning; In-flow; Charging; Queuing Systems. I . I N T RO D U C T I O N Urban transportation systems are facing tremendous chal- lenges no wadays due to the dominant dependenc y and mas- siv e increases on priv ate vehicle ownership, which result in dramatic increases in road congestion, parking demand [1], increased tra vel times [2], and carbon footprint [3] [4]. These challenges can be mitigated with the significant adv ances and gradual maturity of vehicle electrification, autonomous driving (10 million expected vehicles by 2020 [5]), vehicle fast charging infrastructure, and most importantly cyber-physical systems capable of connecting all such components as well as customers to computing engines that can smartly exploit these resources. W ith the rapid development of such cyber -physical systems, it is strongly forecasted that vehicle ownership will significantly decline by 2025, and will be replaced by the no vel concept Autonomous Electric Mobility on-Demand (AEMoD) services [6], [7]. In such system, customers will simply need to press some buttons on an app to promptly get an au- tonomous electric vehicle transporting them door-to-door , with no pick-up/drop-off and driving responsibilities, no dedicated parking needs, no carbon emission, no vehicle insurance and maintenance costs, and extra in-vehicle work/leisure times. AEMoD systems successfully exhibiting all these qualities will significantly prev ail in attracting millions of subscribers across the world, and providing on-demand and hassle-free priv ate urban mobility . Despite the great aspirations for wide AEMoD service deployments by early-to-mid next decade, the stability and timeliness (and thus success) of such service is threatened by two major bottlenecks. First, the expected massiv e demand of AEMoD services will result in excessiv e if not prohibitiv e computational and communication delays if cloud/centralized based approaches are employed for the micro-operation of such systems over an entire city . Moreover , the typical full- battery charging rates of electric vehicles will not be able to cope with the gigantic numbers of vehicles in volv ed in these systems, thus resulting in instabilities and unbounded customer delays. Recent works [9], [10] hav e addressed some important problems related to autonomous mobility on-demand systems but none of them considered the computational architecture for a massiv e demand on such services, nor the vehicle electrification and charging limitations. In our prior work [14], a fog-based architecture [8] with multi-class operation, and possible partial charging was pro- posed to handle these two problems. The fog-based architec- ture distributes the micro-management of AEMoD v ehicles to zone controllers that are close to customers and their most likely allocated vehicles, thus reducing communications and computation loads and delays. The per -zone multi-class oper- ation with partial charging pairs customers with vehicles either having the proper charge or needing a partial charge of their batteries to fulfill the customers’ requested trips. The number of classes is selected to balance the proportions between the customer demands, v ehicle in-flo w , and charging resources of each zone. Decisions on the proportions of vehicles of each class to dispatch or partially/fully charge were optimized in this work to minimize the r esponse time of the system. While the proposed architecture, multi-class operation, and joint dispatching and charging optimization framework in [14] seems very promising, the study assumed a constant vehicle in- flow to each zone. Though this typical by the activ e vehicle in- flow to the system (in-flow of vehicles dropping customers in this zone), the zone demand may require more (less) v ehicles at any gi ven time of the day , which may call for relocating excess vehicles from (to) neighboring zones. One one hand, serving customers within bounded response times can be guaranteed by injecting more vehicles to each zone. On the other hand, one of the key goals of AEMoD systems is to reduce the congestion. Therefore, determining the optimal number of needed vehicles (a.k.a vehicle dimensioning) to stably serve each zone with bounded response time guarantees is very crucial factor in the operation and key goals of AEMoD systems. In addition, such systems need to be resilient and maintain their stability in special conditions like low char ging resources, limited vehicles av ailability , etc. In this paper , we address the above vehicle dimensioning problem with bounded response time guarantees for the fog- based multi-class AEMoD management system proposed in [14]. Using a queuing model representing the multi-class charging and dispatching processes of each zone, we first deriv e the stability conditions and the number of system classes to guarantee the response time bound. Decisions on the proportions of each class vehicles to partially/fully charge, or directly serve customers are then optimized so as to minimize total needed vehicles in-flo w to any giv en zone. Excess waiting times of customers in rare critical e vents, such as limited charging resources and/or limited vehicles av ailabilities, will be also in vestigated. I I . S Y S T E M M O D E L A N D P A R A M E T E R S W e consider one service zone controlled by a fog controller connected to: (1) the service request apps of customers in the zone; (2) the AEMoD vehicles; (3) C rapid charging poles distributed in the service zone and designed for short- term partial charging; and (4) one spacious central charging station designed for long-term full charging. Activ e AEMoD vehicles enter the service in this zone after dropping of f their latest customers in it. Their detection as free vehicles by the zone’ s controller can thus be modeled as a Poisson process with rate λ v . Customers request service from the system according to a Poisson process. Both customers and vehicles are classified into n classes based on an ascending order of their required trip distance and the corresponding suitable SoC for this trip, respectiv ely . From the thinning property of Poisson processes, the arriv al process of Class i customers and vehicles, i ∈ { 0 , . . . , n } , are both independent Poisson processes with rates λ ( i ) c and λ v p i , where p i is the probability that the SoC of an arriving vehicle to the system belongs to Class i . Note that p 0 is the probability that a vehicle arriv es with a depleted battery , and is thus not able to serve customers immediately . Consequently , λ (0) c = 0 as no customer will request a vehicle that cannot tra vel any distance. On the other hand, p n is also equal to 0, because no active v ehicle can arriv e to the system fully charged as it has just finished a prior trip. Upon arri val, each vehicle of Class i , i ∈ { 1 , . . . , n − 1 } , will park anywhere in the zone until it is directed by the fog controller to either: (1) dispatch to serve a customer from Class i with probability q i ; or (2) partially charge up to the SoC of Class i + 1 at any of the C charging poles (whenev er any of them becomes free), with probability q i = 1 − q i , before parking again in waiting to serve a customer from Class i + 1 . As for Class 0 vehicles that are incapable of serving before charging, they will be directed to either fully charge at the central charging station with probability q 0 , or partially charge at one of the C charging points with probability q 0 = 1 − q 0 . In the former and latter cases, the vehicle after charging will wait to serve customers of Class n and 1 , respectively . As widely used in the literature (e.g., [11], [12]), the full charging time of a v ehicle with a depleted battery is assumed to be exponentially distributed with rate µ c , to model the Fig. 1: Joint dispatching and partially/fully charging model, abstracting one service zone of an AEMoD system. random charging duration of different battery sizes. Gi ven a uniform SoC quantization among the n vehicle classes, the partial charging time can then be modeled as an exponential random variable with rate nµ c . Note that the larger rate of the partial charging process is not due to a speed-up in the charging process but rather due to the reduced time of partial charging. The customers belonging to Class i , arriving at rate λ ( i ) c , will be served at a rate of λ ( i ) v , which includes the vehicle in-flow that: (1) arriv ed to the zone with a SoC belonging to Class i and were directed to wait to serve Class i customers; or (2) arri ved to the zone with a SoC belonging to Class i − 1 and were directed to partially char ge to be able to serve Class i customers. Giv en the abov e description and modeling of variables, the entire zone dynamics can thus be modeled by the queuing system depicted in Fig.1. This system includes n M/M/1 queues for the n classes of customer service, one M/M/1 queue for the central char ging station, and one M/M/C queue representing the partial charging process at the C charging points. Our goal in this paper is to minimize the needed rate of vehicle in-flow λ v to the entire zone with respect to the arri val rate of customers in order to guarantee an average response time limit for customers of every class. By response time, we mean the time elapsed between the instant when an arbitrary customer requests a vehicle, and the instant when a vehicle starts moving from its parking or charging spot tow ards this customer . W e will also shade light on the potential dimen- sioning and/or response time relaxation solutions for system resilience in extreme cases of very low energy resources and limited actual vehicle in-flo w . I I I . S Y S T E M S TA B I L I T Y A N D R E S P O N S E T I M E L I M I T C O N D I T I O N S In this section, we first deduce the stability conditions of the proposed system using the basic laws of queuing theory . W e will also derive a lower bound on the number of classes n that fits the customer demands, a verage response time limit, and charging capabilities of any arbitrary service zone. As shown in Fig. 1, each of the n customer classes is served by a separate queue of vehicles having a vehicle in-flow rate λ ( i ) v . Consequently , λ ( i ) v represents the service rate of the customer arriv al in the i th queue. From the aforementioned vehicle dispatching and charging dynamics in Section II, illustrated in Fig. 1, these service rates can be expressed as: λ ( i ) v = λ v ( p i − 1 q i − 1 + p i q i ) , i = 1 , . . . , n − 1 . λ ( n ) v = λ v ( p n − 1 q n − 1 + p 0 q 0 ) (1) Since q i + q i = 1 , q i can be substituted by 1 − q i to get: λ ( i ) v = λ v ( p i − 1 − p i − 1 q i − 1 + p i q i ) , i = 1 , . . . , n − 1 λ ( n ) v = λ v ( p n − 1 − p n − 1 q n − 1 + p 0 q 0 ) (2) From the well-kno wn stability condition of an M/M/1 queue, we must hav e: λ ( i ) v > λ ( i ) c , i = 1 , . . . , n (3) It is also established from M/M/1 queue analysis that the av erage response (i.e., service) time for any customer in the i -th class can be expressed a: 1 λ ( i ) v − λ ( i ) c (4) T o guarantee customers’ satisfaction, the fog controller of each zone must impose an average response time limit T for any class. W e can thus express this av erage response time constraint for the customers of the i -th class as: 1 λ ( i ) v − λ ( i ) c ≤ T (5) which can also be re-written as: λ ( i ) v − λ ( i ) c ≥ 1 T (6) Before reaching the customer service queues, the vehicles will go through a decision step of either going to these queues immediately or partially charging. The stability of the charging queues should be guaranteed in order to ensure the global stability of the entire system at the steady state. From the model described in the previous section, and the well-known stability conditions of M/M/C and M/M/1 queues, we get the following stability constraints on the C charging points and one central charging station queues, respecti vely: n − 1 X i =0 λ v ( p i − p i q i ) < C ( nµ c ) λ v p 0 q 0 < µ c (7) The following lemma sets a lower bound on the av erage vehicle in-flo w rate to the entire service zone to guarantee both its stability and the av erage response time limit fulfillment for all its classes, giv en their demand rates. Lemma 1: For the entire zone stability , and fulfillment of the a verage response time limit for all its classes, the av erage vehicles in-flo w rate must be lower bounded by: λ v ≥ n X i =1 λ ( i ) c + n T (8) Pr oof: The proof of Lemma 1 is in Appendix A in [16]. Furthermore, the following lemma establishes a lower bound on the number of classes n that fits zone’ s customer demands, av erage response time limit, and charging capabilities. Lemma 2: For stablize the zone operation giv en its cus- tomer demands, average response time limit, and charging capabilities, the number of classes n in the zone must obey the following inequality: n ≥ P n i =1 λ ( i ) c − µ c C µ c − 1 /T (9) Pr oof: The proof of Lemma 2 is in Appendix B in [16]. I V . O P T I M A L V E H I C L E D I M E N S I O N I N G A. Pr oblem F ormulation As pre viously mentioned, this paper aims to minimize the av erage vehicle in-flow rate λ v to the entire zone, giv en its charging capacity and customer demand rates, while guaran- teeing an average response time limit for each class customers. Giv en the described system dynamics in Section II and the deriv ed conditions in Section III, the abov e problem can be formulated as a stochastic optimization problem as follows: minimize q 0 ,q 1 ,...,q n − 1 λ v (10a) s.t λ ( i ) c − λ v ( p i − 1 − p i − 1 q i − 1 + p i q i ) + 1 T ≤ 0 , i = 1 , . . . , n − 1 (10b) λ ( n ) c − λ v ( p n − 1 − p n − 1 q n − 1 + p 0 q 0 ) + 1 T ≤ 0 (10c) n − 1 X i =0 λ v ( p i − p i q i ) − C ( nµ c ) < 0 (10d) λ v p 0 q 0 − µ c < 0 (10e) n − 1 X i =0 p i = 1 , 0 ≤ p i ≤ 1 , i = 0 , . . . , n − 1 (10f) 0 ≤ q i ≤ 1 , i = 0 , . . . , n − 1 (10g) λ v ≥ n X i =1 λ ( i ) c + n T (10h) The n constraints in (10b) and (10c) represent the stability and response time limit conditions of the system introduced in (6), after substituting e very λ ( i ) v by its e xpansion form in (2). The constraints in (10d) and (10e) represent the stability conditions for the charging queues. The constraints in (10f) and (10g) are the axiomatic constraints on probabilities (i.e., values being between 0 and 1, and sum equal to 1). Finally , Constraint (10h) is the lower bound on λ v introduced by Lemma (1). The above optimization problem is a quadratic non-con v ex problem with second order differentiable objective and con- straint functions. Usually , the solution obtained by using the Lagrangian and KKT analysis for such non-con ve x problems provides a lower bound on the actual optimal solution. Con- sequently , we propose to solv e the above problem by first finding the solution deriv ed through Lagrangian and KKT analysis, then, if needed, iterativ ely tightening this solution to the feasibility set of the original problem. B. Lower Bound Solution The Lagrangian function associated with the optimization problem in (10) is giv en by the following expression: L ( q , λ v , α , β , γ , ω ) = λ v + α n ( λ v ( p n − 1 q n − 1 − p 0 q 0 − p n − 1 ) + λ ( n ) c + 1 T ) + n − 1 X i =1 α i ( λ v ( p i − 1 q i − 1 − p i q i − p i − 1 ) + λ ( i ) c + 1 T ) + β 0 ( n − 1 X i =0 λ v ( p i − p i q i ) − C nµ c +  0 ) + β 1 ( λ v p 0 q 0 − µ c +  1 ) + n − 1 X i =0 γ i ( q i − 1) − n − 1 X i =0 ω i q i − ω n ( λ v − n X i =1 λ ( i ) c − n T ) , (11) where q is the vector of dispatching decisions (i.e. q = [ q 0 , . . . , q n − 1 ] ), and: • α = [ α i ] , such that α i is the associated Lagrange multiplier to the i -th customer queue inequality . • β = [ β i ] , such that β i is the associated Lagrange multiplier to the i -th charging queue inequality . • γ = [ γ i ] , such that γ i is the associated Lagrange multiplier to the i -th upper bound inequality on q i . • ω = [ ω i ] , such that ω i is the associated Lagrange multiplier to the i -th lower bound inequality on q i and λ v . For more accurate resolutions, two small positive constants  0 and  1 are added to the stability conditions on the charging queues to make them non strict inequalities. Solving the equations giv en by the KKT conditions on the problem equality and inequality constraints, the following theorem illustrates the optimal lower bound solutions of the problem in (10) Theor em 1: The lo wer bound solution of the optimization problem in (10), obtained from Lagrangian and KKT analysis can be expressed as follo ws: λ ∗ v =      P n i =1 λ ( i ) c + n T ω ∗ n 6 = 0 P n i =1 α ∗ i ( λ ( i ) c + 1 T ) − β ∗ 0 ( C nµ c −  0 ) − β ∗ 1 ( µ c −  1 ) ω ∗ n = 0 q ∗ 0 =                0 α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 > 0 1 α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 < 0 p n − 1 q ∗ n − 1 − p n − 1 p 0 + λ ( n ) c + 1 T λ v p 0 α ∗ n 6 = 0 µ c λ ∗ v p 0 β ∗ 1 6 = 0 ζ 0 ( α ∗ , β ∗ , γ ∗ , λ ∗ v , q ∗ ) O therw ise q ∗ i =          0 α ∗ i +1 − α ∗ i − β ∗ 0 > 0 1 α ∗ i +1 − α ∗ i − β ∗ 0 < 0 p i − 1 q ∗ i − 1 − p i − 1 p i + λ ( i ) c + 1 T λ v p i α ∗ i 6 = 0 ζ i ( α ∗ , β ∗ , γ ∗ , λ ∗ v , q ∗ ) O therw ise i = 1 , . . . , n − 1 . (12) where ζ i ( α ∗ , β ∗ , γ ∗ , λ ∗ v , q ∗ ) is the solution that that maximize inf q L ( q , α ∗ , β ∗ , γ ∗ , λ ∗ v ) Pr oof: The proof of Theorem 1 is in Appendix C in [16]. C. Solution T ightening As stated earlier , the closed-form solution deriv ed in the previous section from analyzing the constraints’ KKT con- ditions does not always match with the optimal solution of the original optimization problem, and is sometimes a non- feasible lower bound on our problem. Unfortunately , there is no method to find the exact closed-from solution of non- con ve x optimization. Howe ver , starting from the derived lower bound, we can numerically tighten this solution by toward the feasibility set of the original problem. There are sev eral algorithms to iterativ ely tighten lo wer bound solutions, one of which is the Suggest-and-Impr ove algorithm algorithm proposed in [13] to tighten non-con ve x quadratic problems. W e will thus propose to employ this method whenev er the KKT conditions based solution is not feasible and tightening is required. V . S I M U L A T I O N R E S U L T S In this section, we test both the performance and merits of the proposed dimensioning solution for the considered multi- class AEMoD system. The metric of interest in this study is the optimal vehicle in-flow rate to an arbitrary zone of interest. The performance of the proposed dimensioning solution is tested for two possible SoC distrib utions for in-flow vehicles, namely the decreasing and Gaussian distrib utions. The former distribution better models the more probable active-v ehicle- dominant in-flow scenarios, as such vehicles typically exhibit higher chances of having lower battery charge. The latter distribution models the rarer relocated-vehicle-dominant in- flow scenarios, as such vehicles typically charge for random amounts of times before relocating to the zone of interest. Customers trip distances are always assumed to follow a Gaussian distribution because customers requiring mid-size distances are usually more than those requiring very small and very long distances. For all the performed simulation studies, the full-charging rate of a vehicle is set to µ c = 0 . 033 mins − 1 . Moreov er , for Figures 2, 3, and 4, the number of charging poles C is set to 40. The first important finding of this study is that the obtained solutions using the closed-form expressions in Theorem 1 (i.e., the one derived by applying the KKT conditions) were always feasible solutions to the original problem in (10), for the entire broad range of system parameters employed in our simulations. Thus, the deriv ed closed-form solution is in fact the optimal dimensioning solution for a broad range of system settings, and no tightening is needed. Fig. 2 shows the trade-off relation between the average response time limit, total customer demand rate, and the optimal vehicle in-flo w rate, for both vehicle SoC distrib utions. This curve can be used by the fog controller to get a rough estimate (without exact demand information per class nor optimization of the dispatching and char ging dynamics) on its required in-flow rate (and thus whether it needs extra vehicles or hav e excess vehicles to relocate) for any given customer Fig. 2: Effect of varying the av erage response time limit and total customer demand rate. demand rate and desired response time limit. Fig. 3 illustrates the effect of increasing the number of classes n beyond its lower bound introduced in Lemma 2 for both variable total customer demand rate (while fixing the average response time limits to 5 mins) and variable av erage response time limits (while fixing the total customer demand rate to 5 min − 1 ) in the left and right sub figures, respectiv ely . Both decreasing and Gaussian SoC distributions are considered. In both sub-figure, the lower bound on the number of classes vary depending on the values of the average response time and the total customer demand rate (as shown in Lemma 2), with maximum v alues of 14 and 11 for the employed v alues in the left and right sub-figures, respectively . The results in both figures clearly show that increasing n beyond its lower bound increases the required v ehicle in-flow to the zone. W e thus conclude that the optimal number of classes is the smallest integer v alue satisfying Lemma 2. Fig. 4 compares the performance of our proposed opti- mal vehicle dimensioning scheme with other non-optimized approaches (in which vehicles follo w a fixed dispatch- ing/charging polic y irrespectiv e of the system parameters) for different v alues of total customer demand rate (with T = 5 ) and a verage response time limit (with P n i =1 λ ( i ) c = 5 ). The two non-optimized approaches are the always-charge approach (i.e. q i = 0 ∀ i ) and the equal-split approach (i.e. q i = 0 . 5 ∀ i ). The figure clearly sho ws the superior performance of our deriv ed optimal policy compared to the two non-optimized policies, especially for large total customer demand rates and lower a verage response time limits. For P n i =1 λ ( i ) c = 10 min − 1 in the left subfigure, 36% and 44.4% less vehicle in- flow rates are required compared to always-char ge and equal- split policies, respectively , for the more typical decreasing SoC distribution. These reductions reach 57.6% and 42.5%, respectiv ely for T = 10 min in the right subfigure. The alw ays- charge policy is exhibiting less increase in the required vehicle in-flow rate when the SoC follows a Gaussian distribution. Howe ver , some considerable gains can still be achie ved using our proposed optimized approach in this less frequent SoC 2 4 6 8 10 12 14 Average Response Time Limit (min) 5 6 7 8 9 10 11 12 13 ' 1 n 6 c (i) =5 Optimal n, Gaussian SoC n =11, Gaussian SoC Optimal n, Decreasing SoC n=11, Decreasing SoC 2 4 6 8 10 12 14 Total Customer Demand Rate (min -1 ) 0 5 10 15 20 25 30 Optimal Vehicle in-Flow Rate (min -1 ) T = 5 Optimal n, Gaussian SoC n =14, Gaussian SoC Optimal n, Decreasing SoC n=14, Decreasing SoC Fig. 3: Effect of increasing the number of classes. 2 4 6 8 10 12 14 Total Customer Demand Rate (min -1 ) 0 5 10 15 20 25 30 35 40 45 50 Vehicle in-Flow Rate (min -1 ) T =5 2 4 6 8 10 12 14 Average Response Time Limit (min) 4 6 8 10 12 14 16 18 20 22 ' 1 n 6 c (i) =5 Equal split, Decreasing Optimal 6 v , Decreasing Always charge, Decreasing Optimal 6 v , Gaussian Always charge, Gaussian Fig. 4: Comparison to non-optimized policies. distribution setting. Noting that these gains can be higher in more critical scenarios, the results demonstrate the importance of our proposed scheme in establishing a better engineered and more stable system with less vehicles. Finally , we studied the resilience requirements for our considered model in the critical scenarios of sudden reduction in the number of charging sources within the zone. This reduction may occur due to either natural (e.g., typical failures of one or more stations) or intentional (e.g., a malicious attack on the fog controller blocking its access to these sources). The resilience measures that the fog controller can take in these scenarios is to both notify its customers of a transient increase in the vehicles’ response times giv en the a vailable vehicles in the zone, and request a higher vehicle in-flow rate to gradually restore its original response time limit. Our dev eloped optimization framework in [14] and this paper can easily provide proper numbers for both the abov e two needed actions by the fog controller in charging station outage e vents. The problem of computing the maximum tran- sient response time of the system given the fixed vehicle in- flow rate at failure time was already solved in our pre vious related work [14]. The left subfigure of Fig. 5 depicts the maximum response time v alues of the system for different numbers of av ailable charging poles for a vehicle in-flo w rate λ v = 8 min − 1 and a total customer demand rate of 5 min − 1 . For a Gaussian distribution of v ehicles’ SoC, the response time increases dramatically when the number of charging poles drops below 20. On the other hand, the degradation in response time was much less severe when the SoC of vehicles follows the decreasing distribution. Luckily , the decreasing SoC distribution is the one that is more probable especially at the time just preceding the failure (where most vehicles arriving to the zone are active v ehicles). As for the recovery from this critical scenario and restora- tion of the original response time limit, the proposed di- mensioning framew ork in this paper can be employed to determine the new optimal value of vehicle in-flow rate. The right sub-figure in Fig. 5 depicts the optimal vehicles in- flow λ ∗ v for different values of av ailable charging poles C . In this simulation, the total customer demand rate is set to P n i =1 λ ( i ) c = 8 min − 1 and the av erage response time limit is restored back to T = 10 mins. The figure sho ws that the Gaussian SoC distribution case, which would be luckily the dominant case in this zone after failure time (due to the domination of relocated vehicles called in by the fog controller to recov er from the failure e vent), e xhibit lower need of v ehicle in-flow rate to restore the system conv entional operation. V I . C O N C L U S I O N This paper aimed to formally characterize the optimal vehicle dimensioning for fog-based multi-class AEMoD sys- tems giv en a system-wide a verage response time limit. Using the system’ s queuing model and its stability/response-time constraints, we formulated the optimal vehicle dimensioning problem as a non-con ve x quadratic program over the multi- class dispatching and charging proportions. 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A P P E N D I X A P RO O F O F L E M M A 1 From (2) and (6) we hav e λ ( i ) c + 1 T ≤ λ v ( p i − 1 q i − 1 + p i q i ) , i = 1 , . . . , n − 1 . λ ( n ) c + 1 T ≤ λ v ( p n − 1 q n − 1 + p 0 q 0 ) , i = n (13) The summation of all the inequalities in (13) giv es a new inequality n X i =1 λ ( i ) c + n T ≤ λ v [ n − 1 X i =1 ( p i − 1 q i − 1 + p i q i ) + ( p n − 1 q n − 1 + p 0 q 0 )] (14) n X i =1 λ ( i ) c + n T ≤ λ v [ p 0 q 0 + p 1 q 1 + p 1 q 1 + ... + p n − 1 q n − 1 + p 0 q 0 ] (15) W e have q i + q i so p i q i + p i q i = p i n X i =1 λ ( i ) c + n T ≤ λ v ( p 0 + p 1 + p 2 + ... + p n − 1 ) (16) W e have P n − 1 i =0 p i = 1 so P n i =1 λ ( i ) c + n T ≤ λ v A P P E N D I X B P RO O F O F L E M M A 2 The summation of the inequalities giv en by (7) ∀ i = { 0 , . . . , n } gives the following inequality : λ v n − 1 X i =0 p i − λ v n − 1 X i =0 p i q i + λ v p 0 q 0 < C ( nµ c ) + µ c (17) Since P n − 1 i =0 p i = 1 (because p n = 0 as described in Section 2), we get: λ v − λ v n − 1 X i =1 p i q i < µ c ( C n + 1) (18) In the w orst case, all the vehicles will be directed to partially charge before serving, which means that always q i = 0 . Therefore, we get: C n > λ v µ c − 1 , (19) which can be re-arranged to be: n > λ v C µ c − 1 C (20) From equation (20) and equation (8) we hav e n > λ v C µ c − 1 C ≥ P n i =1 λ ( i ) c + n T C µ c − 1 C (21) By simplifying equation (21) we get n ≥ T P n i =1 λ ( i ) c − µ c T C µ c − 1 (22) A P P E N D I X C P RO O F O F T H E O R E M 1 Applying the KKT conditions to the inequalities constraints of (9), we get: α ∗ i ( λ ∗ v ( p i − 1 q ∗ i − 1 − p i q ∗ i − p i − 1 ) + 1 T + λ ( i ) c ) = 0 i = 1 , . . . , n − 1 . α ∗ n ( λ ∗ v ( p n − 1 q ∗ n − 1 − p 0 q ∗ 0 − p n − 1 ) + 1 T + λ ( n ) c ) = 0 . β ∗ 0 ( n − 1 X i =0 λ v ( p i − p i q ∗ i ) − C ( nµ c ) +  0 ) = 0 . β ∗ 1 ( λ v p 0 q ∗ 0 − µ c +  1 ) = 0 γ ∗ i ( q ∗ i − 1) = 0 , i = 0 , . . . , n − 1 . ω ∗ i q ∗ i = 0 , i = 0 , . . . , n − 1 . ω ∗ n ( λ ∗ v − ( n X i =1 λ ( i ) c + n T )) = 0 . (23) Like wise, applying the KKT conditions to the Lagrangian function in (10), and knowing that the gradient of the La- grangian function goes to 0 at the optimal solution, we get the following set of equalities: λ ∗ v p i ( α ∗ i +1 − α ∗ i − β ∗ 0 ) = ω ∗ i − γ ∗ i , i = 1 , . . . , n − 1 . λ ∗ v p 0 ( α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 ) = ω ∗ 0 − γ ∗ 0 n − 1 X i =1 α ∗ i ( p i − 1 q ∗ i − 1 − p i q ∗ i − p i − 1 ) + α ∗ n ( p n − 1 q ∗ n − 1 − p 0 q ∗ 0 − p n − 1 ) + β ∗ 0 ( n − 1 X i =0 ( p i − p i q ∗ i )) + β ∗ 1 p 0 q ∗ 0 − ω ∗ n + 1 = 0 (24) Knowing that the gradient of the Lagrangian goes to 0 at the optimal solutions, we get the system of equalities giv en by (24). multiplying the first equality in (24) by q ∗ i and the second equality by q ∗ 0 and the third equality by λ ∗ v combined with the equalities giv en by (23) giv es : λ ∗ v p i q ∗ i ( α ∗ i +1 − α ∗ i − β ∗ 0 ) = − γ ∗ i , i = 1 , . . . , n − 1 . λ ∗ v p 0 q ∗ 0 ( α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 ) = − γ ∗ 0 λ ∗ v − n X i =1 α ∗ i ( λ ( i ) c + 1 T ) + β ∗ 0 ( C nµ c −  0 ) + β ∗ 1 ( µ c −  1 ) − ω ∗ n ( n X i =1 λ ( i ) c + n T ) = 0 (25) (25) Inserted in the fifth equality in (23) giv es : λ ∗ v p i ( α ∗ i +1 − α ∗ i − β ∗ 0 )( q ∗ i − 1) q ∗ i = 0 , i = 1 , . . . , n − 1 . λ ∗ v p 0 ( α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 )( q ∗ 0 − 1) q ∗ 0 = 0 λ ∗ v = n X i =1 α ∗ i ( λ ( i ) c + 1 T ) − β ∗ 0 ( C nµ c −  0 ) − β ∗ 1 ( µ c −  1 ) + ω ∗ n ( n X i =1 λ ( i ) c + n T ) (26) From (26) we hav e 0 < q ∗ 0 < 1 only if α ∗ i +1 − α ∗ i − β ∗ 0 = 0 And 0 < q ∗ i < 1 only if α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 = 0 Since 0 ≤ q ∗ i ≤ 1 then these equalities may not always be true if α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 > 0 and we kno w that γ ∗ 0 ≥ 0 then γ ∗ 0 = 0 which giv es q ∗ 0 6 = 1 and q ∗ 0 = 0 . if α ∗ i +1 − α ∗ i − β ∗ 0 > 0 and we know that γ ∗ i ≥ 0 then γ ∗ i = 0 which giv es q ∗ i 6 = 1 and q ∗ i = 0 if α ∗ 1 − α ∗ n − β ∗ 0 + β ∗ 1 < 0 then γ ∗ 0 > 0 (it cannot be 0 because this will contradict with the v alue of q i ), which implies that q ∗ 0 = 1 . if α ∗ i +1 − α ∗ i − β ∗ 0 < 0 then γ ∗ i > 0 (it cannot be 0 because this contradicts with the v alue of q i ), which implies that q ∗ i = 1 W e hav e also from the KKT conditions giv en by equation in in (23) that says either the Lagrangian coefficient is 0 or its the associated inequality is an equality: if β ∗ 1 6 = 0 we hav e q ∗ 0 = µ c λ ∗ v p 0 if α ∗ n 6 = 0 we hav e q ∗ 0 = p n − 1 q ∗ n − 1 − p n − 1 p 0 + λ ( n ) c + 1 T λ v p 0 if α ∗ i 6 = 0 , we hav e q ∗ i = p i − 1 q ∗ i − 1 − p i − 1 p i + λ ( i ) c + 1 T λ v p i for i = 1 , . . . , n − 1 Otherwise by the Lagrangian relaxation: q ∗ i = ζ i ( α ∗ , β ∗ , γ ∗ , λ ∗ v , q ∗ ) for i = 1 , . . . , n − 1 Where ζ i ( α ∗ , β ∗ , γ ∗ , λ ∗ v , q ∗ ) is the solution that that maximize the function inf q L ( q , α ∗ , β ∗ , γ ∗ , λ ∗ v ) Now in order to find the expression of λ ∗ v we first look at the last equation in (23). From there we can say that if ω ∗ n 6 = 0 then λ ∗ v = P n i =1 λ ( i ) c + n T Otherwise from the third equation in (26) if ω ∗ n = 0 then λ ∗ v = P n i =1 α ∗ i ( λ ( i ) c + 1 T ) − β ∗ 0 ( C nµ c −  0 ) − β ∗ 1 ( µ c −  1 )