Vehicular Cloud: Stochastic Analysis of Computing Resources in a Road Segment

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Vehicular Cloud: Stochastic Analysis of Computing Resources in a Road Segment

^†

Tao Zhang

PARADISE Research Lab EECS - University of Ottawa

Ottawa, ON, Canada

tzhan091@uottawa.ca

Robson E. De Grande

Ottawa, ON, Canada

rdgrande@site.uottawa.ca

Azzedine Boukerche

Ottawa, ON, Canada

boukerch@site.uottawa.ca ABSTRACT

Considerable attention has been assigned to Vehicular Cloud towards identifying methods to utilize under-used, available computing and physical resources of vehicles effectively.

Most work on vehicular cloud is so far on the taxonomy definition level, and the dynamically changing amount of available resources characterizes vehicular cloud as more cum- bersome and complex than the traditional Clouds. In this paper, we define and analyze the expected number of vehicles in a roadway segment. These vehicles can serve as the building blocks for the vehicular cloud, enabling a tremen- dously large set of applications that benefit the whole traffic system. To contribute with the analysis, two types of traffic scenarios are considered in this work. We use a macroscopic traffic model to investigate the free-flow traffic, and we utilize the queuing theory to observe the queuing-up traffic.

The average number of vehicles within a roadway segment is calculated using stochastic models. The results show the boundaries on enabling vehicular cloud, allowing to determine a range of parameters for simulating vehicular clouds.

Categories and Subject Descriptors

G.3 [Mathematics of Computing]: Probability and Statis- tics—Stochastic Processes; C.2.1 [Computer-Communica- tion Networks]: Network Architecture and Design—Wire- less Communication

Keywords

Cloud Computing, Vehicular Networks, Stochastic Models, Vehicular Cloud

1. INTRODUCTION

Cloud computing is an internet-based computing technology whereby the large group of remote servers are networked

†This work is partially supported by NSERC, the Canada Research Chair program, MRI/ORF research funds, and the Ontario Distinguished Research Award.

to provide computing resources, storage, and other computer services to customers on demand, like a public utility [8].

The characteristics of a cloud includes virtualization technology, resources on demand, scalability, pay as you go, and Quality of Service (QoS). The cloud computing can provide scalable amount of computing resources and a great deal of Information Technology (IT) services. It has completely re- defined our understanding about the data center and the IT investment of start-up companies. Not surprisingly, cloud computing is becoming the main technological trend among IT companies all over the world [16].

Inspired by the great success of cloud computing and the tremendous technological progress in vehicular networking field, a framework has been proposed in [19] to leverage on- board resources of vehicular fleet. At the same time, this work has introduced the concept of vehicular cloud (VC) to which the next paradigm shift from vehicular networks.

The concept of vehicular cloud is to harness the under- utilized on-board vehicular resources, such as computing power, storage, and network connectivity. Moreover, the ac- cumulated resources can be shared among drivers or rented out to potential customers; this is very much like what has been already implemented for the conventional clouds.

Although the vehicular clouds may be very similar to the conventional clouds from the perspective of utilization, two distinguishing characteristics make it unique and not easy to deal with. Vehicular cloud can provide autonomous cooperation among the vehicular computing resources and offer the decentralized management of those on-board resources of the participating vehicles. In addition, as mentioned before, vehicular cloud has to cope with the continually changing amount of available resources.

In this paper, we take an initial step towards implementing a vehicular cloud on a road segment by modeling and estimating traffic flows. To be more specific, we envision a vehicular cloud involving cars in a road segment. We analyzed two specific traffic scenarios – free flow traffic and queuing-up traffic, as depicted in Figure 1. These two scenarios consist in very common traffic situations nowadays in the urban roadways of metropolitan cities.

In the context of vehicular cloud computing, the first issue is to comprehend the amount of resources that can be harvested from a roadway segment. Specifically, we need to know the number of vehicles that are present in the roadway segment. Since the number of vehicles inside this road segment is dynamic and the travelling time for a specific vehicle varies due to the different speed of vehicles, the amount of available resources is constantly varying. The number of ve-

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Figure 1: Traffic conditidons under the flow of vehicles in road segment

hicles in a road segment, as well as the arrival and departure of vehicles from the segment, can be modelled as a stochastic process.

The structure of the rest of the paper is organized as follows. Section 2 summarizes the concept and recent work of vehicular clouds. Section 3 provides the modeling and analytical results of the study; this section is composed of two subsections, the first subsection focuses on the free-flow traffic scenario, and the second one works on the queuing-up traffic scenario. Finally, Section 4 concludes the paper and presents remarks for future research directions.

2. RELATED WORK

In urban traffic scenarios, usual common vehicles are pres- ently equipped with powerful on-board processing units and several sensing devices, such as GPS device, rear collision sensor, and cameras. Those devices are key in contribut- ing to a safer and less-stressful driving experience, and they are usually under-utilized. Moreover, according to the most recent research work provided by US National Transporta- tion Statistics [22], the registered number of vehicles on the American roadways and streets is approximately 253.6 mil- lion, and this number is increasing steadily. With the extra on-board computing and storage resources, vehicles have the potential to form clouds autonomously using their embed- ded computing capabilities and resolve critical problems in real-time [19].

If vehicle resources can be meaningfully aggregated and utilized properly, the resulting applications show a tremendous positive influence on our society. As such under-utilized vehicular resources containing network connections, com-

puting power, and storage devices can be combined with those of other vehicles. According to a recent work [20], vehicular clouds are economically viable and technologically conceivable, working as a valuable tool to benefit the whole society and introducing the next paradigm shift. Conse- quently, vehicular cloud has evolved from the vehicular networking [9]. The majority of the work related to vehicular cloud discusses about the taxonomy, feasible architectures, possible applications, open issues, and research challenges.

An extensive overview of such studies can be found in [25].

2.1 Vehicular Networks

In the past decades, due to the fast-growing number of vehicles on the road and the urgent demand for the safe and pleasant driving experience, the vehicular networking and its impressive array of potential applications has gained a great amount of popularity. A new sort of network, which evolves from Mobile Ad-hoc Networks (MANET), was developed by researchers called vehicular ad-hoc networks (VANET) [17].

They have been designed to enable the Vehicle to Vehicle (V2V) and Vehicle to roadside Infrastructure (V2I) communications on the streets or highways. The essential function- ality for VANETs consists in providing early notification of possible traffic hazards to the drivers for the sake of avoid- ing any unpleasant driving experience [21]. Each vehicle be- haves like an information collector and also an information distributor. The vehicles near the location of an incident are responsible for gathering the real-time information and passing it to their neighbouring vehicles [24].

2.2 Vehicular Cloud

The concept of vehicular cloud has been first proposed in recent works [1] [19] [20]. The key point of vehicular cloud comprehends the collection, utilization, and alloca- tion of the excessive on-board resources, such as computing capabilities, sensors, storage, and communication resources, in a dynamic group of vehicles under the authorization of the vehicle owners. Combining such resources together and enabling them to provide cloud services to the public benefit both the cloud and owner of vehicles. The services that vehicular clouds can offer are non-trivial and complementary to the conventional cloud computing.

There are many possible applications for the vehicular cloud computing [20]. One of them is the composition and assembly of a datacenter in the parking lot. Since a great number of cars are resting in the garage of a company or a shopping mall for certain amount of time, the on-board storage of these cars can be used as a fundamental resource for forming a datacenter. Another promising application is the dynamic traffic light management system [19]. Traffic jam represent a serious growing issue in metropolitan areas due to the increasing number of vehicles on the road. The drivers who are stuck in congested traffic might be willing to give out their vehicular computing resources so that the traffic management department can perform calculations and run simulations designed to mitigate the traffic congestion by dynamically adjusting the traffic lights.

In [20], many other potential vehicular cloud applications were presented. Those applications can be divided into two major categories [6]. The first group can be classified as a static vehicular cloudlet, which is formed by aggregating the computing capabilities of all the still vehicles resting inside a parking lot, for instance. These kinds of applica-

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tions make a vehicular cloud behave like a traditional cloud.

The other group includes traffic-related applications that are highly dynamic in nature and may be used to assist munic- ipal department of transportation to deal with the traffic congestion or the emergency conditions, such as evacuation plans built in run-time.

For the static vehicular cloud, a research work [2] took an initial step towards implementing a VC and investigated the number of cars that are expected to be present in a long-term parking lot of a typical international airport. The stochastic process with time-varying arrival and departure rates are used to model the number of compute nodes at the datacenter which is formed by the extra computing capabilities of those vehicles. In this particular work, it was deduced a closed form for the expected number of vehicles and its variance in an international airport parking garage.

The time-dependent probability distribution function of the parking lot occupancy and the limiting behaviour of these parameters as the initial conditions fade away. With those parameters, we could have a clear picture of the number of vehicles in the parking lot at any given moment.

At this stage, as far as we know, most work about vehicular cloud is on the taxonomy definition level [25]. Some works focus on the implementation of vehicular cloud in a rather stable environment, such as the parking lot of an international airport or the garage of a large shopping mall.

There are no works on the implementation of vehicular cloud on the road. While a static VC may provide the same services as the traditional cloud facilities, the majority of vehicles spends a substantial amount of time on the road. Fur- thermore, the infrastructure of the vehicular cloud is the combination of the vehicular cloudlet in the parking lot and vehicular cloudlet on the road. In order to provide a more dynamic and broader model, this work presents an initial step towards implementing vehicular cloud in a roadway segment.

2.3 Vehicular Cloud Services

The primary services that vehicular cloud can provide are categorized as Storage as a Service (STaaS), Network as a Service (NaaS), Computation as a Service, and Cooperation as a Service (CaaS) [25].

2.3.1 Storage as a Service

Due to the portable size and cheap price of storage me- dia, it is assumed that vehicles are now equipped with large storage capacity. In this sense, a study [2] investigated the feasibility of building a datacenter in the international airport parking lot by exploiting the under-utilized storage resources in the vehicles. The vehicles need to be plugged into a standard power outlet so as to connect to the vehicular cloud. By giving the air travellers the proper incentives, like free parking or free car-washing, it is anticipated that the vehicle owners are most likely willing to let their vehicles participate in the airport datacenter. In [6], a two-tier data center architecture is proposed for the sake of harnessing the excessive storage in parking lot with a finite capacity. In re- ality, unlike the static storage resources in traditional real datacenter, the number of incoming and outgoing vehicles in a parking lot is a random variable. One of the most popular techniques to deal with this issue is the replication-based fault-tolerant storage [3]. The principle behind this storage demonstrates that the client stores N copies of the original

file at each of the N storage servers. As a result, the original file can be retrieved as long as at least one of the N intact replicas is available.

2.3.2 Network as a Service

The vehicular network is composed by a set of fixed roadside access points (APs) and the mobile vehicular users. The recent works in NaaS area are mainly focusing on utilizing the V2V and V2I connections to transfer data. A study [14]

investigated the performance of downloading content by for- mulating and solving a linear programming problem, given the availability of different data transfer paradigms, which are direct transfer, connected forwarding, and carry-and- forward. Specifically, if a downloader can directly get data from an AP, it is considered a direct-transfer paradigm; if a data packet needs one or more vehicles to create a multi-hop path to reach the downloader, it is defined as connected- forwarding paradigm; if data packets require one or more vehicles to store, carry them, and eventually deliver them to the downloader, it is characterized as carry-and-forward paradigm.

2.3.3 Computation as a Service

Similar to STaaS, the vision of computation-as-a-service is aggregating the excessive computing capabilities of vehicles and presenting it as a new service to authorized customers.

Due to the fact that vehicular cloud computing is a recent technological concept, and it is at its very initial stage, there is not much related work that directly investigates the har- vesting of the computing resources on the vehicles. However, the work done by [2] can also be used as the estimation of computing capabilities inside an airport parking lot. One of the technical challenges of using those on-board computing resources is to find an effective way of migrating the tasks from the outgoing vehicles to the vehicles that are still residing in the parking lot. From the perspective of distributed system, the task migration includes suspending the task, saving the status of computation, finding the target host and migrating jobs.

2.3.4 Cooperation as a Service

A work [11] proposed a Navigator Assisted Vehicular route Optimizer (NAVOPT), which is composed of an on-board navigator and the navigation server based on the cooperation of the vehicular cloud and the traditional cloud. The on-board navigator detects its own geographical coordinates using the internal area map along with GPS and reports it to the server via wireless communication. After collecting enough information from a large number of vehicles in the area, the server is able to construct the traffic load map, calculate the optimal route to the destination for each specific car, and then return the optimized routing strategy to the vehicles.

3. STOCHASTIC MODELING AND ANALYTICAL RESULTS

One of the most important features of a decision support system for the dynamic vehicular cloud is the capability to predict the expected amount of available computational resources on a roadway segment given the random arrivals and departures. In other words, it is indispensable to predict the number of vehicles inside the road segment in the face

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S D

Distance Axis

Time Axis Arrival Reference Point Departure Reference Point

A Roadway Segment

Figure 2: Free-flow Vehicular Traffic over a Roadway Segment [SD]

of the general information about the arrival and departure rates. Two different traffic conditions, free flow traffic and queuing-up traffic, are analyzed in this work for enabling a comprehensive modeling.

Free-flow traffic means that the vehicular density on a roadway segment tend to be low or medium, and consequently, vehicles are more likely to be isolated. In other words, from the stochastic process perspective, the vehicle arrivals to a random geographical reference point follow a Poisson process and they are independent and identically distributed. As a result, the inter-arrival time can be considered as an exponential random process. On the other hand, the queuing-up of traffic usually happens in the rush hours, especially when there are traffic lights or accidents ahead of the traffic flow.

Based on the previously present assumptions and in order to estimate amount of available resources, two traffic scenarios are considered:

• Free Flow Traffic. Based on the free flow model in [10], we analyzed the expected number of vehicles inside the roadway segment and the average residence time for those vehicles. Extensive analytical investiga- tions have been carried out for different scenarios.

• Queuing-up Traffic. The stochastic queuing processes have been used to model the queued traffic.

Base on this model, the average number of vehicles, the mean waiting time in the queue, and the average length of the queue are vital metrics to be used to evaluate traffic density and time of residence.

3.1 Free Flow Traffic

A simple free-flow traffic model has been proposed for vehicular intermittently-connected networks [10]. We mainly use this model as base to derive and carry out our analysis.

As it is mentioned above, this model is based on the traffic theory and can only be used to describe the traffic which presents a low or medium density.

According to the traffic theory in [18], the average vehicle speed observed over a road segment [SD] and depicted in Figure 2 is given by:

V = V¯ max(1 − ρv

ρmax

) (1)

where Vmax is the upper speed limit within this road segment, ρv is the traffic density (veh/meter ) and ρmax is

0 5 10 15 20

0 0.05 0.1 0.15 0.2 0.25

Arrivals in 15 minutes

Probability of Occurence

Mean=0.2 vehicles/minute Mean=0.5 vehicles/minute

Figure 3: Probability of Vehicle Occurrence over Roadway Segment [SD]

a maximum traffic density. In [10], vehicles are assumed to navigate over an uninterrupted roadway segment where there is no intersections, traffic lights, stop signs, and bi- furcations; their speeds are normally distributed within the speed range [Vmin, Vmax], with an average ¯V and a standard deviation σV. Furthermore, vehicles keep the same speeds during the whole navigation period over LSD (length of the road segment). The travel time (i.e. residence time) from S (arrival reference point) to D (departure reference point) of an arbitrary car i with speed vi is described as Ti = ^L_v^SD_i , which is an random variable due to the arbitrary value of the vehicle speed vi. Let FT(τ ) and FV(ν) denote the cumu- lative distribution function (CDF) of the vehicle’s residence time and speed. It can be shown that

FT(τ ) = P [t ≤ τ ] = P [LSD

ν ≤ τ ] = P [ν ≥ LSD

τ ]

= 1 − F^V(LSD

τ )

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Hence, the probability density function (pdf) of vehicle’s residence time is expressed as

fT(t) = dFT(τ )

dτ = M · L^SD t²σV

√2πe

− LSD

t − ¯V σV√

2

!2

(3) Where M is a normalization factor, as described in more details in [10].

According to the realistic data statistics provided in [5], the density of the free flow vehicular traffic can be defined as small or medium. Thus, we can assume that the occurrence of vehicles is a Poisson arrival process, and we can consequently determine that the difference between two consecutive vehicle arrival times, i.e. inter-arrival times, are exponentially distributed. As a result, the expected value of vehicle inter-arrival time is inversely proportional to the vehicle arrival rate µ (veh/s), so the pdf for the inter-arrival time can be written as:

fA(t) = µe⁻^µt (4)

Accordingly, the CDF of the inter-arrival time can be expressed as:

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0 2 4 6 8 10 12 14 16 18 20 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time between Arrivals (minutes)

Probability of exceedance

Mean=0.2 vehicles/minute Mean=0.5 vehicles/minute

Figure 4: Probability of Exceedance over Roadway Segment [SD]

FA(τ ) = P [t ≤ τ ] = Zτ

−∞

fA(t) dτ = 1 − e^−µt (5)

Figure 3 shows the probability of occurrences within 15 minutes when the vehicle arrivals are observed as a Pois- son process. In the situation in which the traffic arrival rate is 0.2 vehicles/minute, with the probability 23%, it is estimated that 2 or 3 vehicles arrived in the road segment in the time period of these 15 minutes. Moreover, if the vehicle arrival rate is 0.5 vehicles/minute, there is a higher chance that 7 vehicles enter this road segment;

the probability of this event is 15%. Figure 4 describes the time intervals between two consecutive vehicle arrivals.

For example, the probability of the inter-arrival time that is more than 4 minutes is 45% when the vehicle arrival rate is 0.2 vehicles/minute. This value decreases significantly to 12% as the vehicle arrival rate goes up to 0.5 vehicles per minute. In addition, the chance of the inter-arrival time that is greater than 12 minutes is zero when the vehicle arrival rate is 0.5 vehicles/minute, which means that the next vehicle can enter the road segment within 12 minutes with a probability of 100%.

Since the vehicle arrival is modelled as a Poisson process, we can use Little’s Formula [12] in queuing theory to estimate the mean number of vehicles inside this roadway segment [SD]. It can be compute as:

E(N ) = µE(T ) = µ Z∞

0

tfT(t) dt

= µ Z∞

0

M · L^SD tσV

√2πe

− LSD

t − ¯V σV√

2

!2

dt

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Based on Formula 6, a thorough investigation of the relation between the experimental parameters and average number of vehicles within the roadway segment [SD] has been conducted. Those parameters include traffic density, the

Table 1: Summary of Experimental Parameters

Name Symbol & Units Default

Values Traffic Density ρ (veh/meter) 0.07 Max Traffic Density ρmax(veh/meter) 0.25 Road Segment Length LSD (meter) 200

Maximum Speed Vmax(m/s) 50

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 5

10 15 20 25

Vehicle Density [Vehicles/meter]

Average number of vehicles

Figure 5: Average Number of Vehicles versus Ve- hicular Density over [SD]

50 150 250 350 450

0 5 10 15 20 25 30 35 40 45 50

Length of Road Segment [meters]

Figure 6: Average Number of Vehicles versus Road Length of [SD]

length of roadway segment, and the maximum speed limit.

The investigation is carried out by performing factor-at-a- time experiments, where one parameter is varied while the other parameters are maintained at their default values. The experimental parameters and their default values are shown in Table 3.1.

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10 15 20 25 30 35 40 45 50 55 60 20

21 22 23 24 25 26 27 28 29 30 31

The maximum speed within the road segment [meters/s]

Road Length=250 m Road Length=300 m Road Length=350 m

Figure 7: Average Number of Vehicles versus Max Speed Limit within [SD]

As the vehicular density increases, the mean number of vehicles inside the roadway segment [SD] is observed to increase, as shown in Figure 5. This relation is presented in this condition because the average vehicle speed ( ¯V ) decreases as ρvbecomes higher, and the traffic flow increases at the same pace. As a result, the vehicle arrivals in a road segment occurs in an increasing frequency.

It is expected that the longer the road is, higher the chances in encountering more vehicles in it. Figure 6 shows the relationship between the mean number of vehicles and the length of roadway segment and it verifies our expecta- tion. It is noticeable the linear relation between number of vehicles and length of segment. Given the same speed distribution, the average vehicle residence time is becoming longer due to the longer road. This means that it will take more time for the vehicles to traverse this road segment.

Therefore, based on the Little’s law, the average number of vehicles inside this road segment increases proportionally.

When the maximum vehicular speed within this segment increases from 10 meters per second (36 km/h) to 60 meters per second (216 km/h), the average number of vehicles increases at the beginning, and then it tends to level off after the maximum speed goes beyond a certain point, as presented in Figure 7. This case is explained as follows.

The increasing maximum speed makes the mean of vehicle speeds observed over this road segment becomes higher, as described in Formula 1, and this leads to the increase of the vehicle arrival rate µ. According to formula 6, the mean vehicle number E(N ) also increases. However, given the fixed road length, as the maximum speed keeps increasing, the mean residence time E(T ) decreases due to the higher speed, as shown in Figure 8. At the end, the increase in arrival rate is offset by the drop of the average residence time.

3.2 Queuing-up Traffic

Traffic congestion is a daily phenomenon in the city due to the increasing number of vehicles on the roads, especially in the rush hours or at the spot where there is an intersection ahead. Figure 9 shows a typical scenario of a vehicle queue.

10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30

The maximum speed within the road segment [meters/s]

Average residence time of vehicles [s]

Road Length=250 m Road Length=300 m Road Length=350 m

Figure 8: Average Residence Time versus Max Speed Limit within [SD]

The Bootleneck Traffic Flow Direction

Figure 9: A illustration of a queue in a road segment When the arrival rate is greater than the departure rate, the queue is bound to form up.

Queuing process [12] is used to model the queuing-up traffic so as to calculate the mean number of vehicles in the queue.

Considering an one-lane roadway segment, we assume that the arrival of vehicles can be represented as a Poisson process with rate λ, so the inter-arrival times are exponential random random variables with mean 1/λ [12], as depicted in Formula 4. Assume that the time it takes for a vehicle to pass by the endpoint of the segment, such as an intersection, a construction site, or any distinguishable point on the segment, follows a general distribution, and we call this period of time as the leaving time (τ ) for a vehicle. The total delay Diof the ith vehicle in this roadway segment is the sum of its waiting time Wiin the queue and the leaving time τi:

Di= Wi+ τi (7)

The steady-state queuing system is considered in this scenario. This means that the state probabilities for this queuing system do not depend on initial conditions. In this case, the average length of the queue maintains a constant value.

We defined ρ as the traffic intensity, which means the expected number of vehicular arrivals E(V ) during the mean leaving time E(τ ). Since the arrival process V is a Poisson with an average of λ arrivals in unit of time, so for a given

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0 0.25 0.5 0.75 1 0

5 10 15 20 25 30

Traffic Intensity

Average number of vehicles in the Queue E(Q)

Fixed leaving time

Exponentially distributed leaving time

Figure 10: Mean Number of Vehicles in a Roadway Segment versus Traffic Intensity

τ , the variable V presents a Poisson distribution with mean λτ . Consequently, it follows that

ρ = E(V ) = λE(τ ) (8)

According to the properties of Poisson distribution, the second moment of V is

E(V²) = V AR(V ) + E²(V ) = λτ + (λτ )² (9) Based on the Formula 8 and 9, the Little’s Law and de- duction in [4] [7], the mean number of vehicles in the road segment (including the vehicle which is departing) can be expressed as

E(N ) = λE(W + τ ) = ρ +λ²σ²τ+ ρ²

2(1 − ρ) (10) where στ² is the standard deviation of the leaving time.

From Formula 7, we can derive the expected value for leaving time E(τ ) = ρ/λ, so the mean waiting time for a vehicle can be expressed as

E(W ) = 1

λ(E(N ) − λE(τ )) = λ²σ²τ+ ρ²

2λ(1 − ρ) (11) and the mean length of the queue is

E(Q) = λE(W ) = λ²σ²τ+ ρ²

2(1 − ρ) (12)

Here we consider two specific cases.

• The leaving time is constant. In this case, we have στ²= 0, so the Formula 12 becomes

E(Q) = ρ²

2(1 − ρ) (13)

• The leaving time is exponentially distributed. The standard variance for the leaving time is σ²τ = E²(τ ) = ρ²/λ², substituting in Formula 12 thus gives

E(Q) = ρ²

1 − ρ (14)

0.2 0.3 0.4 0.5 0.6 0.7

0 5 10 15 20 25 30

Arrival Rate [vehicles/min]

Average Waiting Time in the Queue [min]

Fixed LT (leaving time) (ρ=0.8) Exponentially distributed LT (ρ=0.8) Fixed LT (ρ=0.9)

Exponentially distributed LT (ρ=0.9)

Figure 11: Mean Waiting Time in the Queue Arrival Rate

Note that the average length of queue given in Formula 14 is twice the value in Formula 13. If ρ > 1, which means the vehicle arrival rate is larger than the departure rate, a queue is bound to build up. As a result, we cannot expect to achieve the steady state. It is a must to have ρ ≤ 1 for an equilibrium to be possible. From Formula 12, as well as the two special cases 13 and 14, we can observe that E(Q) = ∞ when ρ = 1. However, if each vehicle inside the road segment has a constant leaving time, and at the same time, the incoming vehicles’ inter-arrival time is also a constant, and it is equal to the leaving time; thus, we should have an equilibrium with ρ = 1. But besides such a special scenario, if we introduce any turbulence to either the inter- arrival time or the leaving time of the system, we must have ρ < 1 to make a steady state possible in the system.

When the steady state is achieved, the average number of queuing vehicles in a roadway segment increases steadily with the traffic intensity, as depicted in Figure 10. It is worth noting that when the traffic intensity is approaching 1, the mean number of queuing vehicles increases to a very large number. This means that there is a serious traffic jam inside this road segment. For example, when ρ = 0.8, the average number in the queue is E(Q) = 0.8²/(1 − 0.8) = 3.2 when the leaving time is exponentially distributed. We can also notice ρ = 0.88, E(Q) = 6.4. Thus, an increase in arrival rate of 0.08/0.8 = 10% leads to a 100% increase in mean number of queuing vehicles.

The average waiting time of a vehicle in the traffic queue is shown in Figure 11. This parameter can be used to evaluate the expiry time of the computing resources of a specific vehicle and the task migration needed to concluded before the resource is unavailable.

4. CONCLUSION AND DIRECTIONS FOR FUTURE WORK

In this paper, we took a very first step towards implementation of a vehicular cloud, namely collecting the computing resources of moving vehicles in a rather dynamic environment, a roadway segment. We analysed two scenarios which

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consist in the free-flow traffic and the queuing-up traffic.

The main objective of this paper is to provide analytical results concerning the availability of computational resources and capabilities in such scenarios. However, unlike a more stable environment, such as the long-term parking lot of an international airport or the garage of the shopping mall, the residence time for a specific vehicle in a road segment is very limited, especially for the free-flow traffic, the travelling time is just a few seconds. As a result, assigning computational tasks to the participating vehicles and migrating the tasks from the leaving vehicles to the vehicles that are inside the targeted road segment are still unresolved issues. However, some previous works about the cluster computing with mobile nodes and message passing interface (MPI) [13] could be the beginning point for tackling this issue.

As future work, for the free-flow traffic, we shall move on to simulations using the third-party simulation tools such as NS2[15] or OMNeT++[23] to determine the process of distribution of the computational tasks to the available vehicles in a roadway segment and migration of the tasks from one vehicle to the other. For the queuing-up traffic, we just discussed the simplest scenario, namely in a one-lane road segment where the equilibrium is achieved. We shall ex- plore scenarios composed of roadway segments with multiple lanes, also covering a non-equilibrium queue.

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