UAV Selection for a UAV-based Integrative IoT Platform

Transcription

1 UAV Selection for a UAV-based Integrative IoT Platform Naser Hossein Motlagh, Milod Bagaa and Tarik Taleb Dep. of Commnications and Networking, School of Elect Eng, Aalto University, Espoo, Finland. s: naser.hossein.motlagh@aalto.fi milod.bagaa@aalto.fi tarik.taleb@aalto.fi Abstract This paper, presents a UAV-based integrative IoT platform that leverages UAVs to deliver different IoT services from height. One of the major tasks of the platform is to select the appropriate UAVs. This selection may be based on different criteria, sch as UAVs eqipment, energy bdget, geographical proximity of the UAV to the area of interest, etc. For the selection mechanism, this paper proposes and formlates two Linear Integer Problem LIP) optimization soltions by aiming at minimizing the energy consmption and shortening the UAV operation time. For the soltions, Energy Aware Selection of UAVs ) and Delay Aware Selection of UAVs ) are evalated throgh simlation. The reslts show that if the objective is the energy, is more efficient than in case of total energy consmption by the UAVs. Additionally, if the time is the objective, has higher performance than in case of operation time. Index Terms Unmanned Aerial Vehicle UAV), UAV/Drone Selection Mechanism, and UAV-based IoT Platform. I. INTRODUCTION For data commnications, Internet of Things IoT) tilize intelligent interfaces for connecting devices, machines, smart objects, smart environments, services, and persons at anytime, anywhere ideally sing any network and any service [1]. These devices not only are installed on the earth, even many of them can be installed as Unmanned Aerial Vehicles UAVs; alternatively known as drones) payload to provide IoT services from the sky whenever needed. UAVs are in different shapes and sizes, sch as fixed wings or mltirotors. Depending on the target application, they are broadly categorized in three classes [2]. Micro and Mini, Tactical, and Strategic UAVs. Micro and Mini UAVs are the smallest ones in size and fly at a low altitde, specifically below 300 meters. They are mostly sed for civil and commercial applications. The applications of UAVs are divers. For instance, they are applied in agricltre, search and resce operations, forestry, ocean and lakes,and oil and gas indstry. For Using UAVs, they mst be eqipped with IoT devices sch as different sensor types and cameras. The deployment of UAV applications leads to very large economic benefits for many companies. Amazon and Finnish Post Ltd, DHL Parcel Copter, and German Cargo Drones are good examples to name as they are sing the UAVs for parcel delivery [3]. There are many applications of UAVs and yet the market of the se of UAVs is in its own starting phase. A recent stdy expects global spending on UAVs to nearly doble from 5.2 in 2013 to 11.6 billion USD by 2023 [4]. In this paper we propose a UAV selection mechanism for a UAV-based integrative IoT platform. To the best of or knowledge, this problem has not been stdied in the literatre. For this reason in the next section we intend to describe or proposed UAV commnication platform. For the UAV selection mechanism, we propose two Linear Integer Problem to deal the problem of UAV selection. The first optimization aims at minimizing the energy consmption as mch as possible in order to handle an event. This soltion aims to ensre a fair energy consmption among different UAVs. Meanwhile, the second optimization problem aims at minimizing the time to handle an event i.e. operation time regardless of the energy consmption. The performance of these proposed soltions are validated throgh simlations. The rest of the paper is organized as follows. Section II describes or proposed commnication network. Section III depicts the UAV selection mechanism and formlation. Section IV provides the optimal soltions for UAV selection and presents the simlation reslts. Finally, Section V concldes the paper. II. UAV COMMUNICATION PLATFORM The basic task of a UAV is collecting data from the remote locations. This data collection system reqires selecting sitable eqipment sch as sensors,and cameras. Choosing the right eqipment depends on the envisioned UAV application. Frthermore, a reliable data transmission system is reqired to share the collected data with the other UAVs and the grond infrastrctre. The selection of the commnication technology shold be based on the region that an event occrs. This is becase for example a UAV may be needed for an operation in a region that there is not any Base Station BS) to spport celllar systems. Therefore there shold be a UAV that can commnicate via Wi-Fi, WiMAX or SATCOM in consonance with the target region and the facilities onboard. Considering that each of the technologies has its own advantages and disadvantages. According to the commnication technology, UAVs can constrct different networking topologies sch as mesh, and mobile/fly ad-hoc manner. Each of the UAVs is planned to accomplish a particlar task. The tasks are assigned by the System Orchestrator SO) which is the manager and the brain of the system architectre. The UAVs are piloted based on predetermined rotes and plans so that they can flfill their

2 tasks. In addition, they are remotely controllable becase they need to be trned on-and-off the sensors or cameras at the right time, in the right place. Figre 1 illstrates or envisioned UAV system architectre. It shows a heterogeneos network of different types of flying UAVs that are spread over a very large zone. UAVs can grop varios clsters in size to accomplish their designated tasks based on the application reqirements and technology limitations. The SO affords to benefit variety of Vale-Added Services VASs). These VASs can be bilt on the top of the deployed UAVs, rather than sing the proprietary ones. Actally, every UAV that eqipped with sitable and remotely controllable devices, like sensors and cameras, can offer these services. A dal-control UAV with a separate control on camera from the GS and UAVs with mlti-fnctional UAVs i.e. sensing the temperatre or hmidity while filming or taking photos) are good examples that provide VASs. A se case scenario is when a UAV is flying to deliver a package to a certain place where its path is predetermined. This UAV can pass above any places like roads, bildings and other predefined paths. Spposing that a transportation company or an individal) needs to know the stats of the traffic in a certain road and the UAV is crrently flying above or near) the intended road. Instead of sing UAVs for the intended tasks), the company/individal can reqest from the owner of this UAV to collect the reqired data, e.g. taking a video from that road. Therefore, a triple benefit is achieved; for the traffic s company/individal as a costmer, for the owner of the UAV, and for the network operator. Considering the VASs, lets assme that mltiple UAVs are flying above a city and each UAV is planned for a specific task. Once in a certain region an event happens and the SO receives a reqest for a UAV to ndertake an emergency task. In this case, the SO needs to decide which UAV is the best choice to assign the task to select it in a smart manner. This decision making and smart UAV selection, first of all, significantly depends on the eqipment onboard, and then the energy level, the time reqired to complete the task, distance and the speed of the UAV. Stdying these parameters is important in flfilling of a task sccessflly and an efficient UAV task management. III. UAV SELECTION The initial consideration for selecting a UAV to do a specific task is sbject to the eqipment reqired by the task/event. Accordingly, the UAV residal energy amont and the energy reqirement of the task is the effective parameter in the UAV selection process. Moreover, the distance to the event location, the UAV speed, the time needed to travel and complete a task; all need to be well investigated by developing optimized algorithms and comptations in order to find an optimal soltion for the best UAV selection. Another consideration is that a UAV selection is application-intended based, i.e. the type of application directly will affect the selection of the UAV. This is becase the application specifies the type of reqired sensors), cameras) or devices) onboard. For example, once the SO receives a reqest that an event has happened in a Fig. 1. UAV network architectre and eqipment-based UAV set selection. region on a certain location. Sch as police needs an emergent photos from the scene of an accident. Moreover, the best UAV selection shold be achieved, in case of minimizing the power consmption and reqired task operation time. For this reason, it is mandatory to seek an optimized way of selecting the best UAV for task accomplishment. A. Problem formlation for selection mechanism For UAV selection mechanism, let denote to a UAV, N be a set of UAVs in the network, E denote to an event, and N i.e. be a UAV in N. lets consider an example where five UAVs are flying and each one carries specific eqipment onboard which pointed by different colors and shapes in figre 1. In the example of the accident and the photo reqired by the police from the scene. The SO shold select a UAV that has the reqired facility i.e. Digital Camera. Therefore, there will be two options UAV1 and UAV3 that can ndertake this task. For another example, once an event occrs and SO receives a reqest for a UAV which shold have Laser Scanner, Low Visibility Camera and Thermal Infrared Sensor. Then, this makes the selection process more complex becase UAVs do not obtain all the eqipment onboard alone and it reqires to apply an efficient selection mechanism. The information from the figre 1 shows that the possible soltion to this shold be a combination of these UAVs. This combination prodces three sets of UAVs as { 1, 5 }, { 2, 5 }, and { 2, 4 } where stands for UAV. After constrcting the possible sets of UAVs, the best set shold be chosen. The reqirement of the best set selection is based on the compting the overall reqired time and energy to accomplish the task. The total time of the UAV operation depends on the travel time of UAV to the event location, sensing and processing and data transmission times. The total needed energy for task completion is compted based on the amont of the energy reqired for the travel, sensing/processing and the transmission energies. The optimal

3 set of the UAVs will be the one that needs minimm amont of time and consmes the lowest amont of energy in order to accomplish and complete the task reqested by the event. B. Energy consmption and reqired time for sensing and processing Any sensor which is applied with UAVs needs its own energy consmption amont and in some cases its own sampling freqency. Generally, it is assmed that a sensor has the following sensing energy consmption. ξ Sensing = V dc I i T i 1) Where, T i is the time needed for obtaining a single sample from the sensor i and I i is the crrent draw of sensor. In order to compte the processing energy of the sensors ξ P rocessing, generally MCU s active and sleep mode for crrents and times are sed as in the following eqation [5]: ξ P rocessing = V dc I mc active T mc active + V dc I mc sleep T mc sleep 2) Then the energy and time of sense-and-process will be: ξsensp rocess = ξ Sensing + ξ P rocessing 3) ΥSenseP rocess = T i + T mc active 4) C. The energy consmption and reqired time for traveling The major energy consmptions by the UAV is the amont of the energy that a UAV tilizes to travel to an event position. This energy is called the Traveling Energy of the UAV,ξ T ravel. In order to compte the travel energy, the travel distance D, E) between UAV and the event position shold be calclated. With having the average velocity V, the traveling time Υ T ravel can be compted. For calclating the travel energy ξ T ravel ; coefficient λ, which represents the amont of energy consmption per one meter shold be compted. λ can be considered as the proportion of the fll battery amont ξ Battery to D Range which is the maximm distance that a UAV can fly. Considering P E = X E, Y E, Z E ) and P = X, Y, Z ); positions of an event and the position of a UAV in 3-D space respectively, and having Υ Endrance, the UAV endrance time. The traveling time and energy of a UAV can be formlated as: D, E) = X X E ) 2 + Y Y E ) 2 + Z Z E ) 2 5) D Range Υ T ravel ξ T ravel = D,E) V 6) = V Υ Endrance 7) λ = ξbattery D Range 8) = λ D, E)9) D. Energy consmption and reqired time for Commnication 1) Commnication Modeling: This section models the commnications between a UAV and an enodeb B. In order to improve the commnication reliability, an atomatic repeat reqest ARQ) scheme is sed for forwarding the information. The ARQ scheme allows resending a packet ntil sccessfl reception at the destination or a maximm nmber of retransmissions M is reached. The nmber of ARQ retransmissions reqired for sccessfl packet reception varies randomly according to the fading channel conditions. Let denotes the transmitting UAV, whereas B refer to the receiving enodeb. The channel gain between and B is referred to as α,b. A Rayleigh block-fading channel is considered, where the channel gain α,b remains constant over one block 1 bt changes independently from one block to another. The received signal y B at a destination node B can be expressed as y B = α,b Px + n B, 10) where P denotes transmission powers of UAV. The symbols transmitted by node is referred to as x. n B is a zero-mean additive white Gassian noise with variance N 0. The term γ,b denotes the instantaneos received signal-to-noise ratio at B given by γ,b = P α 2,B /N 0 [6]. The mean vale of γ,b is denoted as γ,b which can be expressed as γ,b = PE[α2,B ] N 0, 11) where E[α,B 2 ] represents the channel variance, and E[ ] is the expectation operator. Using a distance dependent path loss model, the channel variance can be determined as [7] ) β E[α 2 x,y ] = D0, 12) D,B where D,B refers to the distance between nodes and B, D 0 denotes a reference distance typically set to 1 m, and β denotes the path loss exponent. In or physical model, we take into accont the effects of both path loss and fast fading, while the impact of shadowing is neglected. Theorem 1. For any UAV N fails to transmit its packet to an enodeb B iff SNR,B falls below a threshold γ th. This event is known as an otage event and occrs with a probability P,B which can be expressed as P,B = 1 exp γ th γ,b ). 13) Proof. In this appendix, we derive the proof for the otage probability between and B in the network. By definition, the link B is in otage if SNR,B falls below a threshold level γ th [6]. This event occrs with a probability P,B. In order to determine an expression for the otage probability P,B, we need first to compte the CDF of SNR,B. First, we recall that the expression of the SNR is given by SNR,B = γ,b, 14) 1 A block corresponds to the time dration necessary to send one packet.

4 where γ,b is exponential random variables with mean vales γ,b. The probability density fnctions PDFs) of γ,b is: p γ,b z) = 1 exp z ), 15) γ,b γ,b The random variable γ,b follows an exponential distribtion and its CDF is given by F γ,b z) = 1 exp z ), 16) γ,b P,B = P ) ) SNR,B γ th = P γ,b γ th = F γ,b z) = 1 exp γ th γ,b ). 17) 2) Commnication time modeling: The UAVs be eqipped with a bffer to store the packets before their transmission. We assme that UAV has K packets that represent the sensed information abot an event E. This section focses on the analysis of the average delay of transmission Υ T ransmit of sensed data from to an enodeb B. We do not take into accont neither the time reqired for sensing and processing nor the time reqired for traveling from the event E to B. represents the sojorn time of a packet before its transmission to B. Formally, Υ T ransmit = K T T ransmit. A sccessfl reception of a packet at enodeb B occrs after a random nmber of retransmissions. To qantify the delay associated with the retransmission events, we measre the Let T T ransmit average sojorn time T T ransmit of a packet in the bffer of, which is defined as the average time elapsed from the starting of its transmission ntil its sccessfl reception. The packet s sojorn time in the bffer can be evalated sing the Pollaczek- Khinchin eqation as [8], T T ransmit = EN,B )T F, 18) where T F is the time reqired for a single transmission of a given packet and EN,B ) is the average nmber of retransmissions for the packets sent by. For the ARQ scheme, the packet is retransmitted ntil sccessfl reception at the B or a maximm nmber of retransmissions M is reached. In case of reception failre after M retransmissions the packet is discarded. The nmber of retransmissions N,B varies randomly according to the position of UAV and the conditions of the fading channel between the and B. The average nmber of retransmissions EN,B ) can be expressed as [9], EN,B ) = 1 + = P F 1,..., F m ) = 1 + P,B ) m P,B ) m = 1 ) M P,B, 19) 1 P m=0,b where P F 1,..., F m ) is the probability of a reception failre at the 1,..., mth retransmissions. Since the channel realizations in each transmission are independent identically distribted, the event of reception failre at each step are independent and have eqal probabilities, ths P F 1,..., F m ) = P,B ) m. From 18 and 19, we have: = K T F EN,B ) 1 ) M P,B Υ T ransmit 1 P,B. 20) 3) Energy consmption modeling in commnication: This section stdies the energy consmption at UAV. Lets assme that UAV has K packets that represent the sensed information abot an event E. The fact that the nmber of retransmissions varies depending on the channel conditions makes the consmed power a random variable. This section stdies the average consmed power and then dedce from that the average consmed energy. The average consmed power P for the ARQ scheme can be determined as P = P P S 1 ) + 2P P F 1, S 2 ) M 1)P P F 1,..., S M 1 ) + MP x P F 1,..., F M 1 ) ) ) = P 1 + P F 1,..., F m ) = P 1 + P,B ) m = P ET,B ) = P 1 P,B ) M 1 P,B, 21) where the term P stands for the power per retransmission at a given. We denote by P S 1 ) the probability of sccessfl reception at B of the first transmission, while P F 1,..., S M 1 ) refers to the probability of a reception failre in the 1st, 2nd,..., M 2)th retransmissions and a sccessfl reception at the M 1)th retransmission. If the packet is sccessflly received after the first transmission this event occrs with a probability P S 1 )), the amont of consmed power wold be eqal to P. If the packet is received correctly after two retransmissions the probability of this event is P F 1, S 2 )), the amont of consmed power wold be eqal to 2P. The consmed power wold be eqal to MP if the 1st,..., M 1)th retransmissions fails the probability of this event is P F 1,..., F M 1 )). The average consmed power is obtained by smming p all the possible vales of consmed power weighted by their respective probability of occrrence. The reslt in 21) shows that we can express the average consmed power as the prodct of two terms: the power per retransmission P and the average nmber of retransmissions ET,B ). The average consmed energy Φ,B of one packet can be obtained as Φ,B = P T F P S 1 ) + 2P T F P F 1, S 2 ) M 1)P T F P F 1,..., S M 1 ) + MP T F P F 1,..., F M 1 ) ) = P T F 1 + P,B ) m = P T F ET,B ) = P T F. 22) The average consmed energy ξ T ransmit to transmit the whole data K packets) can be obtained as ξ T ransmit = K Φ,B = K P T F. IV. OPTIMAL SOLUTIONS FOR UAV SELECTION Let we denote by ξ T ot and Υ T ot, the total energy and the total time reqired by a UAV to accomplish a mission for handling an event E. Formally, ξ T ot Υ T ot = ξ T ravel = Υ T ravel SenseP rocess + ξ SenseP rocess + Υ + ξ T ransmit, 23) + Υ T ransmit, 24)

5 Here, we propose two Linear Integer Problem to deal the problem of UAV selection. In the first optimization problem, we aim to minimize as mch as possible the energy consmption to handle the event E. This soltion aims to ensre a fair energy consmption among different UAVs. Meanwhile, the second optimization problem aims at minimizing the time to handle an event regardless of the energy consmption. Before starting the selection process, the eligible UAVs shold be selected. Let N denotes the eligible UAVs in the network. Formally, N is defined as the UAVs that satisfy the following conditions: Power constraint: The selected UAVs shold have the enogh power to ndertake a task, i.e. ξ Battery > ξ T ot ; Eqipment constraint: This constraint ensres that the eligible UAV can attend the event E and it has the reqired sensors for the operation. Formally this constraint is defined by the following conditions: A UAV has the ability of flying that allows it to reach the altitde of the event E. Formally, mh Z E ; mh is the maximm altitde that a UAV can fly. A UAV has the reqired sensors to handle the event E. Formally, S S E. S E is the sensor/s reqired by the event and S refers to UAV Sensor/s onboard. A. Optimization of energy consmption Let X be a boolean decision variable that represents if a UAV N shold be selected to handle the event E. { 1 If is selected to handle the event E X = 25) 0 Otherwise This soltion aims to minimize as mch as possible the energy consmption by selecting the minimm nmber of UAVs in the network while the time delay do not exceed a predefined threshold Υ th. This soltion is formlated throgh the following Linear Integer Problem: min ξ T ot N X s. t. s S E : X 1 N s S N : Υ T ot X Υ th N : X {0, 1} 26) Where, the first constraint ensres that the selected UAVs have the reqired sensors to deal with the event E. The second constraint ensres that the time latency of each selected UAV shold not exceed the threshold Υ th. The last constraint ensres that X is a boolean decision variable. B. Optimization of operation time Let X be a boolean decision variable that represents if a UAV N shold be selected to handle the event E. { 1 If is selected to handle the event E X = 27) 0 Otherwise This soltion aims to minimize as mch as possible the time response of UAVs while the residal energy of these UAVs shold exceed a predefined threshold ξ th. This soltion is formlated throgh the following Linear Integer Problem: min max Υ T N ot X s. t. s S E : X 1 N s S N : ξ Battery - ξ T ot ) X ξ th N : X {0, 1} 28) Where, the first constraint ensres that the selected UAVs have the reqired sensors to deal with the event E. The second constraint ensres that the residal energy in each selected UAV shold be higher than the ξ th. The last constraint ensres that X is a boolean decision variable. C. Simlation reslts For or proposed optimal soltions, we developed a simlator sing Python and implemented throgh Grobi optimization tool. In the simlations, we call the soltions as Energy Aware Selection of UAVs ) and Delay Aware Selection of UAVs ). In the simlation reslts, each plotted point represents the average of 200 times of exections. The plots are presented with 95 % confidence interval. The applied algorithm evalates the performance of the UAV selection in terms of the nmber of UAVs and the size of UAVs flying area. For both cases the total energy consmption and the operation time is considered. It is assmed that there is only one event has happened which reqires 6 different type of sensors and UAVs have variant nmber of sensors onboard. In the simlations, performances of and are evalated two times. First by varying nmber of UAVs with fixed area size eqal to sqare meters. Second evalation made by varying area size with fixed nmber of UAVs eqal to ) Performances evalation by varying nmber of UAVs: For this evalation, with the objective of total energy consmption, Fig.2a) represents the nmber of UAVs in the network and the total energy consmption by the attending UAVs. It shows increasing nmber of UAVs in the network and with this increment, the availability of the reqired nmber of sensors for the event increases. This means that a selection, i.e. set of UAVs jointly can accomplish the task. By increasing the nmber of the UAVs the reqired amont of energy for the operation is decreasing. For example a set with 250 UAVs, reqires 100 mah per UAV to do the task in the manner while it is more efficient and consmes less power nearly 200 mah per UAV) compared to the. With the objective of operation time, Fig.2b) explains that by increasing the nmber of the UAVs in the network, the reqired operation time for participating in an even is considerably decreasing. In addition, in the comparison shows mch better performance than the optimization in terms of operation time. 2) Performances evalation by increasing the size of flying Area: For this evalation, with the objective of total energy consmption, Fig.3a) demonstrates the size of UAVs flying area with the high performance of toward in terms of total energy consmption. This is becase, for example if

6 Total energy consmption mah) Nmber of UAVs Operation time s) Nmber of UAVs a) The energy consmption b) The operation time Fig. 2. Performances evalation by varying the nmber of UAVs Total energy consmption mah) Operation time s) Area size km Area size km 2 a) The energy consmption b) The operation time Fig. 3. Performances evalation by increasing the size of flying Area all the UAVs are flying in an area of 25 Km 2, in order to do the task they consme 125 mah per UAV in manner while they mst consme 300 mah per UAV in manner. This illstrates that has more efficiency than where the size of operation area is increasing. With the objective of operation time, Fig.3b) explains that by increasing the flying area of UAVs the reqired operation time for participating in an event is slightly increasing and in the comparison, shows better performance than the optimization. V. CONCLUSION In near ftre billions of IoT devices sch as sensors, and cameras that are installed on the earth or are on the fly like UAVs onboard will be connected. The UAVs can constrct a commnication network in an ad-hoc manner in a way that some of them are on the fly and others are ready to fly when needed. Each of these UAVs can be assigned to accomplish a particlar task. Employing UAVs can offer a variety of Vale-Added IoT Services. These VASs can be bilt on the top of the deployed UAVs, rather than sing the proprietary ones. Selecting the best UAV from a set of flying ones, is sbject to the capability of the UAV in terms of obtaining the reqired eqipment onboard and the crrent residal energy of battery of the UAV. In this paper, we referred to or envisioned UAV commnication platform and we discssed the different components needed to bild a realistic UAV-based IoT system. For an efficient UAV selection mechanism we provided the formlation that reslts in the best UAV set selection from among the all sets. For the soltions we proposed two optimization problems sing Linear Integer Problem to select the optimal set of UAVs. For the soltions and the reslt of or optimization work, we presented Energy Aware Selection of UAVs ) and Delay Aware Selection of UAVs ) that evalated throgh the simlations. For these soltions UAV selection is simlated in terms of the nmber of the UAVs and the size of the UAVs flying area. For both cases the total energy consmption and the operation time is considered. The reslt of this simlation depicts that; in UAV selection if the operation time is the target to achieve, which aims to minimize as mch as possible the response times of UAVs shold be applied. Additionally, in UAV selection if the objective is the energy consmption by UAVs; that aims to minimize as mch as possible the energy consmption shold be sed. REFERENCES [1] V. Ovidi and F. Peter, Internet of Things: Converging Technologies for Smart Environments and Integrated Ecosystems. River Pblishers, [2] C. Gowei, D. Jroge, and S. Lakma, A Srvey of Small-Scale Unmanned Aerial Vehicles: Recent Advances and Ftre Development Trends, Unmanned Systems, vol. 2, no. 2, p. 125, Feb [3] A. Ricardo. 2016) Delivery drones via air. [Online]. Available: [4] F. Phil. 2015) Aerospace and Defence Market Intelligence Analysis and Forcasts, Missile and UAV Market Forecasts. [Online]. Available: [5] A. R. Mohammad and D. Simon, Energy-Efficient Sensing in Wireless Sensor Networks Using Compressed Sensing, Sensors, vol. 14, no. 2, pp , Feb [6] M. K. Simon and M.-S. Aloini, Digital Commnication over Fading Channels, 2nd ed. New Jersey, USA: John Wiley & Sons, [7] A. F. Molisch, Wireless Commnications. Wiley-IEEE Press, [8] W. Chan, T.-C. L, and R.-J. Chen, Pollaczek-Khinchin formla for the M/G/1 qee in discrete time with vacations, IEE Proceedings - Compters and Digital Techniqes, vol. 144, no. 4, pp , Jl [9] A. Chelli, E. Zedini, M.-S. Aloini, J. R. Barry, and M. Pätzold, Performance and delay analysis of hybrid ARQ with incremental redndancy over doble Rayleigh fading channels, IEEE Trans. on Wireless Commn., vol. 13, no. 11, pp , Nov