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Delay Analysis of Unsaturated HeterogeneousOmnidirectional-Directional Small Cell Wireless
Networks: The Case of RF-VLC CoexistenceSihua Shao and Abdallah Khreishah
Abstract—The coexistence of omnidirectional small cell(OSC), such as RF small cells, and directional small cell(DSC), such as visible-light communication (VLC) cells, isinvestigated. The delay of two cases of such heterogeneousnetworks is evaluated. In the first case, resource allocatedOSCs, such as RF femtocell, are considered. In the secondcase, contention-based OSCs, such as WiFi access point, arestudied. For each case, two configurations are evaluated.In the first configuration, the non-aggregated scenario,any request is either allocated to OSC or DSC. Whilein the second configuration, the aggregated scenario, eachrequest is split into two pieces, one is forwarded to OSCand the other is forwarded to DSC. For the first case,under Poisson request arrival process and exponentialdistribution of request size, the optimal traffic allocationratio is derived for the non-aggregated scenario and itis mathematically proved that the aggregated scenarioprovides lower minimum average system delay than thatof the non-aggregated scenario. For the second case, theaverage system delay is derived for both non-aggregatedand aggregated scenarios, and extensive simulation resultsimply that, under certain conditions, the non-aggregatedscenario outperforms the aggregated scenario due to theoverhead caused by contention.
Keywords—Heterogeneous network (HetNet), delay, om-nidirectional small cell (OSC), directional small cell (DSC),RF femtocall, WiFi, visible light communications (VLC), linkaggregation.
I. INTRODUCTION
Demand for ubiquitous wireless connectivity con-tinues to grow due to the trend towards an alwayson culture, broad interest in mobile multimedia,and advancement towards the Internet of things.This demand stems from a multifaceted growthin the number of networked devices and the per-device data usage from novel applications (e.g.,HD video, augmented reality, and cloud-based ser-vices). Forecasts from Cisco show Internet video
Sihua Shao and Abdallah Khreishah are with the Departmentof Electrical and Computer Engineering, New Jersey Institute ofTechnology, emails: [email protected] and [email protected]
accounting for 80% of all consumer Internet trafficby 2019 [1] while Qualcomm and Ericsson expectbetween 25 and 50 billion connected devices by2020 [2], [3]. Next generation, or 5G, wirelessnetworks will be challenged to provide the capacityneeded to meet this growing demand. Compared topeak performance goals of previous generations, 5Ggoals include increasing the expected performanceacross non-uniform geographic traffic distributions.In particular, additional capacity is needed in denseurban environments and indoor environments whereapproximately 70% of IP-traffic occurs [4].
Heterogeneous wireless network, as a method toincorporate different access technologies, containsthe potential capabilities of improving the efficiencyof spectral resource utilization. Traffic offloading toomnidirectional small cells (OSCs), such as RF fem-tocells and WiFi WLANs, has already become an es-tablished technique for adding capacity to dense en-vironments where macrocells are overloaded. Ultra-dense distributed directional small cells (DSCs),deployed in indoor environments, can supplementOSCs in areas like apartment complexes, coffeeshops, and office spaces where device density anddata demand are at their highest. These DSCs canbe implemented by technologies like microwave [5],mmwave [6] and optical wireless. Optical wireless(OW) communication - specifically visible lightcommunication (VLC) or LiFi [7] - is a direc-tional communication technology that has gainedinterest within the research community in recentyears. As an excellent candidate for 5G wirelesscommunication, VLC provides ultra wide bandwidthand efficient energy utilization [8]. However, theweaknesses of VLC is the vulnerability to obstacleswhen compared to the omnidirectional RF commu-nication.
In this work, we consider two cases of het-erogenous OSC-DSC networks. One case is thecoexistence of resource allocated OSCs (RAOSCs)
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and DSCs. A typical application of RAOSC is theRF femtocells [9], which are owned/controlled bya global entity (i.e., service provider). Therefore,interference can be mitigated in the provisioningprocess and multiple adjacent RF femtocells canperform downlink data transmission simultaneouslywithout contention. This non-contention issue willbe further discussed in Section II. The other case isthe heterogenous network incorporating contention-based OSCs (CBOSCs) and DSCs. In contrast toRAOSC, CBOSC (such as WiFi AP) is purchasedby local entities (i.e. home/business owners) anddeployed in an ad-hoc manner such that interferenceis not planned. Particularly, WiFi networks employsthe Carrier Sense Multiple Access with CollisionAvoidance (CSMA/CA) protocols to schedule thecontention process. DSCs have a large reuse factorsuch that the spectrum reuse can be easily imple-mented even in an indoor environment. Without theloss of generality, we use OSC and DSC notationsinstead of RF and VLC in the following description.
Many current research efforts have been paidtowards developing heterogeneous networks incor-porating both OSC and DSC. A protocol, con-sidering OFDMA, vertical handover (VHO) andhorizontal handover (HHO) mechanisms for mobileterminals (MTs) to enable the mobility of usersamong different VLC APs and OFDMA system, isproposed in [10]. The authors define a new metric,called spatial density, to evaluate the capacity of theheterogeneous network under the assumption of theHomogenous Poisson Point Process (HPPP) distri-bution of MTs. In [11], load balancing for hybridVLC and WiFi system is optimized by both central-ized and distributed resource-allocation algorithmswhile achieving proportional fairness. In [12], dif-ferent RF-VLC heterogeneous network topologies,such as symmetric non-interfering, symmetric withinterference and asymmetric, are briefly discussed.In [13], taking the advantage of wide coverage ofRF and spatially reuse efficiency of VLC, a hybridRF and VLC system is proposed to improve per useraverage and outage throughput.
Regarding the bandwidth aggregation, a thoroughsurvey of approaches in heterogeneous wireless net-works has been presented in [14]. The challengesand open research issues in the design of bandwidthaggregation system, ranging from MAC layer toapplication layer, have been investigated in detail.The benefits of bandwidth aggregation includes in-
creased throughput, improved packet delivery, loadbalancing and seamless connectivity. This work alsovalidates the feasibility of the heterogeneous OSC-DSC networks proposed here based on bandwidthaggregation. In [15], users connect to WiFi and VLCsimultaneously. A parallel transmission MAC (PT-MAC) protocol containing CSMA/CA algorithmand the concept of parallel transmission are pro-posed. This protocol supports fairness among usersin the hybrid VLC and WLAN network.
The above-mentioned works, which are primarilysimulation-based studies, do not provide system-level implementation of the WiFi-LiFi systems. Inour previous work [16]–[18], an aggregated WiFi-VLC system is presented and implemented usingWiFi/VLC equipment and Linux Bonding driver.The realized WiFi-LiFi system aggregates a singleWiFi link and a single VLC link, and providesimproved throughput. This paper theoretically inves-tigates system delay, a critical QoS metric especiallyfor multimedia applications [19]. Here, system delayis defined as the amount of the time from the instantthe request arrives at the AP to the instant that itsuccessfully departs from the AP.
In [19], delay modeling of a hybrid WiFi-VLCsystem has been investigated. Each WiFi and VLCqueue is observed as an M/D/1 queue, and the ca-pacities with respect to the unstable delay points ofWiFi only, asymmetric WiFi-VLC and hybrid WiFi-VLC systems are compared. An analytic model forevaluating the queueing delays and channel accesstimes at nodes in 802.11 based WiFi networks ispresented in [20]. The model provides closed formsolutions for obtaining the values of the delay andqueue length. This is done by modeling each node asa discrete time G/G/1 queue. However, these worksdo not investigate the delay modeling of a systemwith bandwidth aggregation. In other words, mostof the existing heterogeneous works only study thenetworks without bandwidth aggregation (i.e. onerequest is either forwarded to one access technologyor the other).
This paper characterizes the system delay oftwo cases of heterogeneous OSC-DSC wirelessnetworks: (i) RAOSC-DSC; (ii) CBOSC-DSC. Foreach case, two configurations are taken into con-sideration. One of them is based on bandwidthaggregation and the other is not. The potential gainin terms of the minimum average system delaythrough aggregating the bandwidth of OSC and DSC
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is also evaluated. To the best of our knowledge,this work is the first to quantify the system per-formance of aggregation with respect to minimumaverage system delay. Note that investigating thedelay performance of a heterogeneous system whenaggregation is considered, is our major contribution,which differs from other existing works. The maincontributions of this work include the following:(i) for the heterogeneous RAOSC-DSC wirelessnetwork, a generalized characterization of the sys-tem without bandwidth aggregation is derived interms of the optimal ratio of traffic allocation andthe minimum average system delay and a near-optimal characterization of the minimum averagesystem delay of the system that utilizes bandwidthaggregation is proposed; (ii) for the heterogeneousRAOSC-DSC wireless network, it is also theoreti-cally proved that the minimum average system delayof the system based on bandwidth aggregation islower when compared to that of the system withoutbandwidth aggregation; (iii) for the heterogeneousCBOSC-DSC wireless network, the average systemdelay is derived for both the system without band-width aggregation and the system with bandwidthaggregation; (iv) for the heterogeneous CBOSC-DSC wireless network, extensive simulations arealso conducted to indicate that under certain con-ditions, the system without bandwidth aggregationoutperforms the system with bandwidth aggregationin terms of minimum average system delay.
II. SYSTEM MODEL
A. ParametersA recent measurement study [21] on traces of
3785 smart phone users from 145 countries over afour-month period shows that the ratio of downloadtraffic to its upload traffic is 20:1. Therefore, inthis paper, we investigate the downlink system delayof two cases of heterogeneous OSC-DSC wirelessaccess networks:• Case 1: heterogeneous RAOSC-DSC network,• Case 2: heterogeneous CBOSC-DSC network.Fig. 1 illustrates the network architecture for case
1. In the system model suggested, there are oneRAOSC AP and N1 DSC APs. Since OSC APs donot contend with each other, under homogeneoustraffic distribution, the delay analysis of a singleRAOSC AP can be easily extended to that of mul-tiple RAOSC APs. Due to the fact that the DSCs
Central
Coordinator
Internet
Wired Interconnection
Average requests arrival rate Average request size
RAOSC bandwidth DSC bandwidth
DSC wireless channel
RAOSC
AP
Fig. 1: Heterogeneous RAOSC-DSC network archi-tecture
Router and CBOSC
Access Point
Internet
Wired Interconnection
Average requests arrival rate Average request size
CBOSC bandwidth DSC bandwidth
Router and CBOSC
Access Point
Internet
Wired Interconnection
Interference
DSC wireless channel DSC wireless channel
Fig. 2: Heterogeneous CBOSC-DSC network archi-tecture
have a large reuse factor [22], it is rational to assumethat all the DSC links can be active simultaneouslywith negligible interference among them. Under thehomogeneous traffic assumption, the traffic assignedto different DSC APs is evenly distributed. The re-quests arrival process to the central coordinator is aPoisson process [20], [23] with rate λ1. One requesthere means one download session (e.g. a photo, awebpage, a video) from the Internet. For prioritysystem [24], where each session forms a flow witha certain priority level and packets of lower prioritystart transmission only if no higher priority packet iswaiting, Poisson arrival process is applicable due tothe independency among a large number of arrivalof requests. Since the requests are from differentindependent sources, it is assumed that the size ofeach request is exponentially distributed with meanµ1. The downlink capacities of the RAOSC and theDSC are Bw
1 and Bv1 , respectively, where Bw
1 < Bv1 .
Fig. 2 illustrates the network architecture for case2. In this case, there are M CBOSC APs and N2
DSC APs, where N2 > M . All of the M CBOSCAPs are located in a single contention domain.
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The MAC scheme considered is IEEE 802.11 [25],which is implemented by using a Distributed Coor-dination Function based on the CSMA/CA protocol.The RTS/CTS exchange scheme, which is utilizedto address the “hidden node” problem, is also takeninto account. The 802.11 configurations will be de-scribed in details in Section IV. The blockage prop-erty of DSC is modeled as a successful transmissionprobability Psucc for each request. The whole requestwill be retransmitted once the transmission fails. TheAck-enabled mechanism [26] for DSC is considered.Under the homogeneous traffic assumption, the traf-fic assigned to different CBOSC and DSC APs areevenly distributed. The requests arrival process toeach AP is a Poisson process with rate λ2/M . Thesize of each request is exponentially distributed withmean µ2. The downlink capacities of the CBOSCand the DSC are Bw
2 and Bv2 , respectively.
For two cases of heterogeneous OSC-DSC wire-less access networks, the system delay D perfor-mance is studied for two configurations: i) non-aggregated scenario and ii) aggregated scenario.In the non-aggregated scenario, any request is ei-ther allocated to the RAOSC/CBOSC or the DSC.In the aggregated scenario, each request is splitinto two pieces. One of them is forwarded to theRAOSC/CBOSC while the other is forwarded to oneof the DSC APs. In the paper, one request meansone download session (e.g. a photo, a webpage, avideo) from the Internet. For the aggregated sce-nario, assume one request consists of 1000 packets,to implement aggregation, these 1000 packets aresplit into two sets - one contains β portion of packetsand the other contains the remaining (1−β) portionof packets. To aggregate the bandwidth of OSC andDSC, the β portion of packets will be transmittedthrough the OSC channel and simultaneously the(1 − β) portion of packets will be sent via theDSC channel. To implement such a heterogeneoussystem, one central coordinator is needed. Thecentral coordinator is an additional device encom-passing multiple functionalities, such as collectingthe location and channel information of all APsand user terminals, computing the optimal trafficallocation ratio, and forwarding the data traffic todifferent APs. Most of the hybrid RF-VLC papers[13], [18], [19], [27], [28] have utilized the centralcoordinator in the system for performing the trafficallocation functionality. Also the cost of the centralcoordinator is usually cheap, such as banana pi [29].
TABLE I: The definition of some of the symbols
λ1(λ2) Total request arrival rate for the heterogeneous RAOSC-DSC network(the heterogeneous CBOSC-DSC network)
µ1(µ2) Mean size of request for the heterogeneous RAOSC-DSCnetwork(the heterogeneous CBOSC-DSC network)
Bw1 (Bw
2 ) RAOSC(CBOSC) bandwidthBv
1 (Bv2 ) Case 1 DSC(Case 2 DSC) bandwidth
M The number of CBOSC APsN1(N2) The number of Case 1 DSC(Case 2 DSC) APsPsucc The successful transmission rate for DSC linksα1(α2) The percentage of traffic allocated to RAOSC(CBOSC)β1(β2) The proportion of the size of each request assigned to
RAOSC(CBOSC)
As a result, the system delay of each request is themaximum of i) time spent by the piece of request inRAOSC/CBOSC and ii) time spent by the piece ofrequest in DSC. The system delay of the requestsin RAOSC, CBOSC and DSC are represented byDRAOSC , DCBOSC and DDSC , respectively. Newmetrics α1(α2) and β1(β2) are defined for two cases,to represent the traffic allocation ratio and requestsplitting ratio for non-aggregated and aggregatedscenarios, respectively. These four factors will bediscussed in detail in Section III and Section IV.The main notations are summarized in Table. I.
B. Overview of Typical Omnidirectional Non-Contention and Contention Wireless Networks
As we discussed earlier, a typical example of om-nidirectional non-contention wireless network is theRF femtocell network. RF femtocell is a small andlow-power cellular base station, typically designedfor coverage and capacity improvement. One of themost critical issues from deploying RF femtocellsis the potential interference among femtocells andmacrocells [30]. However, femtocells can incor-porate interference mitigation techniques-detectingmacrocells, adjusting power and scrambling codesaccordingly [31] to eliminate the potential interfer-ence. The interference management among neigh-boring femtocells and among femtocells and macro-cells are also investigated in [32]. Clustering of fem-tocells [33], [34], fractional frequency reuse (FFR)and resource partitioning [35], [36], and cognitiveapproaches [37] can be employed to mitigate theinter-femtocells interference. Since femtocells aredeployed by service provider, who has the priorityof manipulating the frequency, power, and locationof all the femtocells, the above-mentioned interfer-ence mitigation techniques can be applied without
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contention issue. With interference issue solved, theneighboring RF femtocells can perform downlinkdata transmission at the same time without worryingabout the contention process even at the cell edge.
For omnidirectional contention-based wirelessnetwork, a typical example is WiFi network. Sinceeach WiFi AP is normally deployed independentlywithout coordination with the neighboring WiFiAPs, the interference among WiFi APs will in-evitably trigger the contention process when theadjacent WiFi APs perform the downlink data trans-mission simultaneously. The CSMA/CA based MACprotocol of IEEE 802.11 [25] is designed to mitigatethe collisions due to multiple WiFi APs transmittingon a shared channel. In a WiFi network employingCSMA/CA MAC protocol, each WiFi AP with apacket to transmit will first sense the channel duringa Distributed Inter-frame Space (DIFS) to decidewhether it is idle or busy. If the channel is idle,the WiFi AP proceeds with the transmission. If thechannel is busy, the WiFi AP defers the transmissionuntil the channel becomes idle. The WiFi AP theninitializes its backoff timer with a randomly chosenbackoff period and decrements this timer every timeit senses the channel to be idle. The timer stopsdecreasing once the channel becomes busy and thedecrementing process will be restarted again afterDIFS idle sensing. The WiFi AP attempts to transmitonce the timer reaches zero. The backoff mechanismand the definition of contention window will bediscussed later in Section IV.
III. SYSTEM DELAY ANALYSIS FORHETEROGENEOUS RAOSC-DSC NETWORK
This section presents the mathematical deriva-tion of the minimum average system delay of thenon-aggregated scenario for heterogeneous RAOSC-DSC networks when negligible blockage rate ofDSC is considered. It provides a theoretical proofthat under this case the performance of the aggre-gated scenario is always better than that of the non-aggregated scenario in terms of the minimum aver-age system delay. For the evaluation of the minimumaverage system delay of the aggregated scenario, anefficient solution is proposed. This solution is shownto achieve less than 3% close to the optimal solution.The comparison between the empirical results of theaggregated scenario and the delay performance ofthe non-aggregated scenario is also presented. In the
M/M/1
M/M/1
M/M/1
Fig. 3: Queuing model representing the non-aggregated system model for heterogenous RAOSC-DSC networks
end, when non-negligible blockage rate of DSC isassumed, we use simulation results to evaluate theminimum average system delay of the aggregatedand non-aggregated scenarios.
A. The Non-aggregated Scenario
Let α1 denote the percentage of requests allocatedto RAOSC. The non-aggregated scenario can berepresented by the queuing model shown in Fig. 3.Due to the assumption that requests are randomlyforwarded to RAOSC and DSC, the requests arrivalto each queue is still a Poisson process. Requestsarrive to RAOSC and DSC queues with mean ratesα1λ1 and (1− α1)λ1/N1, respectively. The averageservice time of RAOSC and DSC queue are expo-nentially distributed with means Bw
1 /µ1 and Bv1/µ1,
respectively. Thus, each RAOSC and DSC queue ischaracterized by the M/M/1 queuing model.
Theorem 1: In the non-aggregated system model,the minimum average system delay is
Dmin non agg ={µ1N1
Bv1N1−λ1µ1
, if Bv1N1
λ1µ1(1−
√γN1) ≥ 1
λ1µ1(1+N1)−Bv1N1(1−
√γN1)2
λ1[Bv1N1(γ+1)−λ1µ1]
, otherwise
Proof: The optimization problem for minimiz-ing the average system delay is formulated as fol-lows:
Objective: min α1DRAOSC + (1− α1)DDSC
s.t. 0 ≤ α1 ≤ 1
α1λ1 < Bw1 /µ1 (1)
(1− α1)λ1/N1 < Bv1/µ1 (2)
6
In order to find the candidate minimum points,the average system delay as a function is describedas follows:
D(α1) = α1DRAOSC + (1− α1)DDSC
=α1
Bw1 /µ1 − α1λ1
+1− α1
Bv1/µ1 − (1− α1)λ1/N1
D(α1) is continuous in (1 −Bv
1N1/(λ1µ1), Bw1 /(λ1µ1)). From constraints
(1) and (2), we have 1 − Bv1N1/(λ1µ1) < 0 and
Bw1 /(λ1µ1) > 1. Hence, D(α1) is continuous in
[0,1]. The derivative of D(α1) is
D′(α1) =aα2
1 + bα1 + c
f 2(α1), where
a = λ21(B
w1 −Bv
1N21 ),
b =2λ1B
w1 (B
v1N1 − λ1µ1 +Bv
1N21 )
µ1
,
c = [Bw1 ((B
v1)
2N21 − 2λ1µ1B
v1N1 + λ2
1µ21
−Bw1 B
v1N
21 )]/µ
21,
f(α1) =√µ1(−λ1α1 +
Bw1
µ1
)(λ1α1
N1
+Bv
1
µ1
− λ1
N1
).
It is found that f 2(α1) = 0 when α1 is in [0,1].Since a < 0 and b2 − 4ac > 0, D′(α1) has two zeropoints α1(1) and α1(2)
α1(1) =λ1µ1
√γ/(Bv
1N1) +√γ(√γN1 − 1)
λ1µ1(√γ +
√N1)/(Bv
1N1)(3)
α1(2) =
√γ[1−Bv
1N1(√γN1 + 1)/(λ1µ1)]
√γ −
√N1
(4)
α1(2)− α1(1) =2√γN1[1−Bv
1N1(γ + 1)/(λ1µ1)]
γ −N1
(5)
where γ = Bw1 /(B
v1N1) and γ < 1. In (3), the
numerator is less than λ1µ1/(Bv1N1) and the de-
nominator is greater than λ1µ1/(Bv1N1). Thus, this
proves α1(1) < 1. In (4), the numerator and thedenominator are both less than zero. This provesthat α1(2) > 0. In (5), since the numerator anddenominator are both less than zero, α1(2) is greaterthan α1(1). This means that i) D′(α1) < 0 whenα1 < α1(1) or α1 > α1(2); ii) D′(α1) > 0 whenα1(1) < α1 < α1(2).
The discussion is divided into four cases: i) 0 <α1(1) < 1 and 0 < α1(2) < 1; ii) α1(1) ≤ 0 and0 < α1(2) < 1; iii) 0 < α1(1) < 1 and α1(2) ≥ 1;iv) α1(1) ≤ 0 and α1(2) ≥ 1. In case i) and iii),
M/M/1
M/M/1
M/M/1
Fig. 4: Queuing model representing the aggregatedsystem model for heterogeneous RAOSC-DSC net-works
for the first case, D′(α1) is negative in the range of[0, α1(1)) and (α1(2), 1], and positive in the rangeof (α1(1), α1(2)). Also because D(0) < D(1), thusDmin(α1) = D(α1(1)). For the third case, D′(α1)is negative in the range of [0, α1(1)) and positivein the range of (α1(1), 1]. Therefore, Dmin(α1) =D(α1(1)). In case ii) and iv), Dmin(α1) = D(0)because D(0) < D(1). After substituting α1 = 0and α1 = α1(1) into D(α1), it is found that
D(0) =µ1N1
Bv1N1 − λ1µ1
and
D(α1(1)) =λ)1µ1(1 +N1)−Bv
1N1(1−√γN1)
2
λ1[Bv1N1(γ + 1)− λ1µ1]
Note that Dmin non agg = D(α1(1)) iff α1(1) > 0.It means that Bv
1N1
λ1µ1(1−
√γN1) < 1.
B. The Aggregated Scenario
Let β1 denote the proportion of the size of eachrequest that is allocated to the RAOSC. The aggre-gated scenario can be represented by the queuingmodel shown in Fig. 4. Assuming that the requestsarrival are randomly and evenly distributed to eachDSC queue, the requests arrival process to eachDSC queue is still a Poisson process. The averagerequests arrival rates for RAOSC and DSC areλ1 and λ1/N1, respectively. The average servingrates of RAOSC and DSC are Bw
1 /(β1µ1) andBv
1/[(1 − β1)µ1], respectively. Similar to the non-aggregated scenario, each RAOSC and DSC queuecan be characterized by the M/M/1 queuing model.The objective of the optimization problem can beexpressed as minimizing E[max(DRAOSC , DDSC)].
7
CC
CC
Fig. 5: Requests distribution in the aggregated sce-nario for N1 = 1 and N1 > 1
Fig. 5 represents the requests distribution toRAOSC and DSC queues for N1 = 1 and N1 > 1.In Fig. 5, it can be seen that when N1 = 1, thedelay of the DSC queue is fully correlated to that ofthe RAOSC queue. Therefore, achieving the objec-tive value of minimizing E[max(DRAOSC , DDSC)]is equivalent to obtaining the optimal β1 fromE[DRAOSC ] = E[DDSC ]. However, when N1 >1, the RAOSC queue contains different coloredpieces of request, which are split from the re-quests flowing to different DSC APs. Each colorrepresents a data stream destined to one DSCAP. The arrival times and the sizes of differentcolored pieces of request are independent whilethose of the same colored pieces of request arecompletely correlated. Specifically, due to the ex-istence of yellow and green pieces of request (inFig. 5) in the RAOSC queue, the departure timesof the red pieces of request in the RAOSC queueand the DSC queue are neither independent norcompletely correlated. Hence, the complexity ofcomputing the optimal β1 is severely exacerbated.Instead of searching for the optimal β1 by minimiz-ing E[max(DRAOSC , DDSC)], the objective is sim-plified as minimizing max(E[DRAOSC ], E[DDSC ]).For instance, let us assume that the delaysof three pieces of request in RAOSC are 1,2 and 3 seconds respectively, and the delaysof the corresponding three pieces of requestin DSC are 2 seconds for all. As such, theobjective value of E[max(DRAOSC , DDSC)] willbe 2.33 seconds while the objective value ofmax(E[DRAOSC ], E[DDSC ]) will be 2 seconds,
which provides an underestimation of the trafficload. When the RAOSC queue is overwhelmed,approximated E[DRAOSC ] will be lower than thereal average request delay and vice versa. The errorvalue has been further validated not to exceed 3%by the simulation results. To determine the approxi-mated value of the optimal β1 from the objective ofminimizing max(E[DRAOSC ], E[DDSC ]), we makeE[DRAOSC ] = E[DDSC ]. Therefore, the approxi-mated value of β1 is, β1 = (−b−
√b2 − 4ac)/(2a),
where a = λ1µ1(1 − 1/N1), b = −[Bw1 + Bv
1 +λ1µ1(1− 1/N1)], and c = Bw
1 .
By simulating the aggregated scenario with theapproximated β1, the percentages of additional delaycaused by approximation are shown in Fig. 6. Thevalues of the λ1, µ1, B
w1 , B
v1 are initially set as 0.5/s,
90 Mb, 50 Mpbs, 100 Mbps, respectively. In eachplot, one of these four parameters is varied whilekeeping the other three fixed to the initial values.With N1 varied from 1 to 10, it is noticed thatthe percentage of the maximum additional delay is2.7%, which is less than 3%. Figs. 6 (a)-(c), showthat, as λ1, µ1 and Bw
1 increase, the percentage ofthe additional delay decreases initially and increasesafter reaching the minimum level. However, in Fig. 6(d), the percentage of the delay penalty does notchange much. Figs. 6 (a)-(c) show that the percent-age of additional delay has the minimum valueswhen λ1 ≈ 0.33, µ1 ≈ 58 and Bw
1 ≈ 70, respec-tively. When λ1 < 0.33, µ1 < 58 and Bw
1 > 70, theapproximation approach overestimates the conges-tion level of RAOSC and causes additional trafficload allocated to DSC, and vice versa. Note thatwhen N1 = 1, the approximated solution proposedhere will lead to the exact minimum average systemdelay of the aggregated scenario because the delayof requests at each queue are fully correlated. Theexplicit additional delay values are shown in Fig. 7.
C. Theoretical Analysis
Theorem 2: Under our heterogeneous RAOSC-DSC network model, the aggregated scenario hasa lower minimum average system delay than that ofthe non-aggregated scenario.
Proof: The average system delays of the non-
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The
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aggregated and the aggregated scenarios are
E[Dnon agg] =α1
Bw1 /µ1 − α1λ1
+1− α1
Bv1/µ1 − (1− α1)λ1/N1
E[Dagg] =E[max(DRAOSC , DDSC)]
=E[DRAOSC ] + E[DDSC ]
− E[min(DRAOSC , DDSC)]
Note that, for aggregated scenario,
E[DRAOSC ] =1
Bw1
β1µ1− λ1
=β1
Bw1
µ1− β1λ1
E[DDSC ] =1
Bv1
(1−β1)µ1− λ1
N1
=1− β1
Bv1
µ1− (1−β1)λ1
N1
When α1 = β1, since E[min(DRAOSC , DDSC)] isgreater than zero, we always have E[Dnon agg] >
E[Dagg]. Therefore, the minimum average systemdelay of the aggregated scenario is lower than thatof the non-aggregated scenario.
D. Empirical AnalysisWhen applying the approximation method, the
following question should be addressed: is the re-sulting minimum average system delay with approx-imated β1 of the aggregated scenario still lowerthan that of the non-aggregated scenario? To fur-ther investigate the comparison between the non-aggregated and the aggregated scenarios, the an-alytical results obtained when applying the non-aggregated scenario are compared with the simu-lation results obtained when applying the approx-imated aggregated scenario. The ratio of the ap-proximated minimum average system delay of the
9
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Fig. 8: The ratio of the approximated minimum average system delay of the aggregated scenario to theminimum average system delay of the non-aggregated scenario in terms of (a) λ1; (b) µ1; (c) Bw
1 ; (d) Bv1 ,
with N1 varied from 1 to 10
aggregated scenario to the minimum average systemdelay of the non-aggregated scenario is used todemonstrate the viability of the approximation ap-proach. Fig. 8 illustrates the comparison. The valuesof λ1, µ1, B
w1 , B
v1 and N1 are the same as those in
Fig. 6. As such, based on the simulation parameters,the approximated minimum average system delay ofthe aggregated scenario is at least 16% lower thanthat of the non-aggregated scenario. The aggrega-tion has diminishing gains over the non-aggregatedscenario as the number of DSC APs increases andthe ratio of RAOSC bandwidth to DSC bandwidthdecreases. This is due to the additional RAOSC ca-pacity which leads to decreasing the effect per DSCAP. Besides, the benefit of aggregating RAOSC andDSC becomes less evident as λ1 and µ1 increases.This is because increasing traffic load reduces theeffect of efficient bandwidth utilization provided byaggregation.
E. Extension to non-negligible blockage rate ofDSC
As it will be discussed in the next section, thequeuing model of DSC would be changed to M/G/1if non-zero blockage rate is considered. As a result,it would be very difficult to mathematically derivethe minimum average system delay of the non-aggregated scheme for heterogeneous RAOSC-DSCnetworks and also very complicated to theoreticallycompare the performance of the aggregated schemeand that of the non-aggregated scheme in terms ofthe minimum average system delay. Note that the
mathematical derivation and theoretical comparisonare both performed in the first case (i.e. RAOSC-DSC) when negligible blockage rate is considered.
To evaluate the RAOSC-DSC case when non-negligible blockage rate of DSC is assumed, weperform simulations with the settings similar to thatof the negligible blockage rate case, but change theblockage rate from 0 to 0.1 and 0.2. The simulationresults of RAOSC-DSC case are shown in Fig. 9and Fig. 10, respectively. Comparing the results inFig. 9 and Fig. 10 to the results in Fig. 8, weobserve that the variation trend of the ratio of theminimum average system delay of the aggregationscenario to that of the non-aggregation scenario arevery similar. As it is expected, the only differenceis that when non-zero blockage rate is consideredfor the DSC channels, the benefit of performingaggregation increases. This is consistent with thesimulation results in the Fig. 8. As the bandwidthof DSC decreases, which is similar to increase theblockage rate of DSC channel, the gain of perform-ing aggregation is enhancing. Therefore, the sameconclusion when blockage is not considered can bedrawn when blockage is considered.
IV. SYSTEM DELAY ANALYSIS FORHETEROGENEOUS CBOSC-DSC NETWORK
In this section, we first model the system delay ofthe non-aggregated and the aggregated scenarios forheterogeneous CBOSC-DSC networks. To validateour analytical model, we conduct extensive simula-tions based on the system model presented in Sec-tion II. We also observe from the simulation results
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Fig. 9: For the case of RAOSC-DSC, when blockage rate of DSC is 0.1, the ratio of the minimum averagesystem delay of the aggregated scenario to that of the non-aggregated scenario in terms of (a) λ1; (b) µ1;(c) Bw
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Fig. 10: For the case of RAOSC-DSC, when blockage rate of DSC is 0.2, the ratio of the minimum averagesystem delay of the aggregated scenario to that of the non-aggregated scenario in terms of (a) λ1; (b) µ1;(c) Bw
1 ; (d) Bv1 , with N1 varied from 1 to 10
that, under certain conditions, the non-aggregatedscenario outperforms the aggregated one in terms ofminimum average system delay. This is due to thefact that the delay penalty introduced by aggregationwhen contention and backoff mechanism is utilizedsurpasses the benefit of splitting the request.
A. The Non-aggregated scenarioLet α2 denote the percentage of requests allo-
cated to CBOSC. The non-aggregated scenario canbe represented by the queuing model in Fig. 11.Similar to the analysis for heterogeneous RAOSC-DSC networks, the request arrival process to eachqueue is still a Poisson process. However, sincethe contention and backoff of 802.11 protocols areconsidered when modeling the CBOSC network,
M/G/1
M/G/1
M/G/1
M/G/1
Fig. 11: Queuing model representing the non-aggregated system model for heterogeneousCBOSC-DSC networks
11
the service time of each CBOSC queue Tw(α2)depends on the traffic load allocated to CBOSC.Also, for DSC queues, due to the consideration ofthe blockage, the distribution of the service timeof each request T v is not memoryless. Therefore,the M/G/1 queuing model is utilized to characterizeeach CBOSC and DSC queue. In order to fullycharacterize the delay of the resulting M/G/1 model,we need to derive the expectation and the secondmoment of the service time of the resulting M/G/1model.
The minimum and maximum contention windowsize associated with backoffs are denoted by CWmin
and CWmax, respectively. In 802.11 protocol, mis defined as m = log2(CWmax/CWmin). For in-stance, CWmin = 16 slots and CWmax = 1024slots, and thus m = 6 for 802.11n protocol. Inthe following analysis, since RTS/CTS exchange isconsidered, we denote the probability that an RTStransmission results in a collision by p. Followingthe same approach in [20] ((5) in [20]), the averagenumber of backoff slots experienced by a request ata CBOSC AP can be expressed as
W =1− p− p(2p)m
1− 2p
CWmin
2. (6)
Denote the duration consumed by a collision byTc = DIFS+σRTS , where Distributed Inter-FrameSpace (DIFS) is utilized to sense the idle channeland σRTS = lRTS/B
w2 is the transmission delay of
an RTS packet. Given the average request arrival rateas α2λ2
Mand the average time to transmit a request in
CBOSC queue as µ2
Bw2
, the collision probability canbe expressed as follows according to [20] ((11) in[20])
p = 1−(1−
α2λ2
M[1 + 1
W( µ2
Bw2+ Tc
p2(1−p)
)]
1− α2λ2
M(M − 1)[ µ2
Bw2+ Tc
p2(1−p)
]
)M−1
. (7)
By substituting (6) into (7), the collision rate p canbe obtained by numerical methods.
Denote the queue utilization rate of each CBOSCAP as ρ, then according to [20] ((10) in [20]), wehave
ρ =
α2λ2
M[ µ2
Bw2+ Tc
p2(1−p)
+ W ]
1− α2λ2
M(M − 1)[ µ2
Bw2+ Tc
p2(1−p)
].
Next, we start deriving the probability densityfunction (pdf) of the request service time, which is
from the instant that the request reaches the headto the queue to the instant that the request departsfrom the queue. The pdf of the backoff slots (BO),following a successful transmission of a request ata CBOSC AP, is represented by
P [BO = i] =ρ(1− p)U1,CWmin(i) + p(1− p)
× [U1,CWmin∗ U1,2CWmin
(i)] + ...
+ (p)m(1− p)[U1,CWmin∗ U1,2CWmin
∗ ... ∗ U1,2mCWmin](i)],
where Ua,b denotes the pdf of a uniform distributionbetween a and b, and ∗ represents the convolutionoperation.
To evaluate the portion of service time resultedfrom the successful transmissions and collisions ofthe contending CBOSC APs, we denote q as theprobability that one of the remaining M−1 CBOSCAPs attempts to transmit in a given slot, and qc asthe probability that a collision occurs in a slot giventhat at least one of the M−1 CBOSC APs attemptsto transmit in that slot. According to [20] ((13) and(15) in [20]), we have
q = 1− (1− ρ
W)M−1,
and
qc =1− (1− ρ
W)M−1 − (M−1)ρ
W(1− ρ
W)M−1
1− (1− ρW)M−1
.
Assume that in the i backoff slots, j slots arefollowed by transmission attempts of the other M−1CBOSC APs and k out of j slots are followed bycollisions, then j−k slots are followed by successfultransmissions of the M − 1 CBOSC APs. Sincethe summation of j − k i.i.d. exponential randomvariables (i.e. transmission time of a request µ2
Bw2
) isa gamma random variable, the contribution of j− ksuccessful transmissions to the service time can beexpressed as a gamma distribution
l(j−k)(x) =1
(j − k − 1)! (Bw
2
µ2)j−k
xj−k−1e−µ2x
Bw2 .
Then the pdf of the channel access delay experi-enced by a request is given by
P [Y = s] =∞∑i
i∑j
j∑k
l(j−k)(x)
(i
j
)qi(1− q)i−j
×(j
k
)qkc (1− qc)
j−kP [BO = i]I(s),
(8)
12
where(ij
)qi(1− q)i−j represents the probability that
j out of i slots are followed by transmission attemptfrom the M − 1 CBOSC APs,
(jk
)qkc (1 − qc)
j−k
represents the probability that k out of j slots arefollowed by collisions, and I(s) is an indicatorfunction which equals 1 when s = x+ i+ kTc and0 otherwise.
Denote the moment generating function (mgf) ofthe channel access delay by MY (t), the mgf ofthe total service time MR(t), including the channelaccess delay and request transmission time, is givenby
MR(t) = MY (t)(1− t(Bw
2
µ2
)−1)−1,
where (1 − t(Bw
2
µ2)−1)−1 represents the mgf of an
exponential random variable with mean µ2
Bw2
. Thenthe second moment and the mean of the total servicetime Tw can be obtained by differentiating MR(t)with respect to t and setting t = 0 as follows
E[(Tw)2] =d2MR(t)
dt2(0), E[Tw] =
dMR(t)
dt(0).
According to Pollaczek-Khinchine formula, theexpected system delay of CBOSC queues is givenby
E[DCBOSC ] =α2λ2
ME[(Tw)2]
2(1− ρ)+ E[Tw].
For DSC queues, in order to fully characterize theaverage system delay of requests, we need to derivethe expectation and the second moment of the ser-vice time of the resulting M/G/1 model. Recall thatthe probability of successful transmission is denotedby Psucc and packet drop due to buffer limitation isnot considered. Although in some cases, a packetmay be dropped after a certain number of unsuccess-ful retransmissions, the error caused by this infiniteextension is negligible since Psucc(1−Psucc)
n−1 → 0as n increases. Therefore, the expected service timeof a request in DSC queues is
E[T v] =µ2
Bv2
[Psucc + 2Psucc(1− Psucc) + ...
+ nPsucc(1− Psucc)n−1 + ...]
=µ2
Bv2Psucc
.
Suppose a request’s transmission time is v andthe number of transmission attempts is u, then thetotal service time of this request is uv. Thus, the
M/G/1
M/G/1
M/G/1
M/G/1
Fig. 12: Queuing model representing the aggregatedsystem model for heterogeneous CBOSC-DSC net-works
second moment of the service time of a request inDSC queues is
E[(T v)2] =∞∑v
∞∑u
Bv2
µ2
e−Bv
2µ2
vPsucc
(1− Psucc)u−1(uv)2.
According to Pollaczek-Khinchine formula, theexpected system delay of DSC queues is given by
E[DDSC ] =
(1−α2)λ2
N2E[(T v)2]
2(1− (1−α2)λ2
N2E[T v])
+ E[T v].
Since α2 portion of the requests are allocated toCBOSC networks and 1 − α2 portion of requestsare allocated to DSC networks, the average systemdelay of the heterogeneous CBOSC-DSC networksbased on the non-aggregated scenario is given by
Dnon agg = α2E[DCBOSC ] + (1− α2)E[DDSC ].
B. The aggregated scenarioLet β2 denote the proportion of the size of each
request that is allocated to the CBOSC. The aggre-gated scenario can be represented by the queuingmodel in Fig. 12. Similar to the non-aggregatedscenario for heterogeneous CBOSC-DSC networks,the request arrival process of each CBOSC or DSCqueue can be described by a Poisson process, andthe distribution of service time are not memorylessfor both CBOSC and DSC queues. Therefore, weuse the M/G/1 queuing model to characterize thesystem delay of each CBOSC and DSC queue.
For the derivation of the system delay for the ag-gregated scenario, we only describe the parameters
13
p, ρ, l(j−k)(x), MR(t), E[DCBOSC ], E[T v], E[(T v)2]and E[DDSC ] with different expressions when com-paring them to those of the non-aggregated scenario.Given the average request arrival rate of CBOSCqueues as λ2
Mand the average time to transmit a
request in CBOSC queue as β2µ2
Bw2
, the collisionprobability, queue utilization and the contribution ofj−k successful transmissions to the service time canbe expressed as follows
p = 1−(1−
λ2
M[1 + 1
W(β2µ2
Bw2
+ Tcp
2(1−p))]
1− λ2
M(M − 1)[β2µ2
Bw2
+ Tcp
2(1−p)]
)M−1
, (9)
ρ =
λ2
M[β2µ2
Bw2
+ Tcp
2(1−p)+ W ]
1− λ2
M(M − 1)[β2µ2
Bw2
+ Tcp
2(1−p)], (10)
l(j−k)(x) =1
(j − k − 1)! (Bw
2
β2µ2)j−k
xj−k−1e−β2µ2x
Bw2 .
(11)
Substitute (9), (10) and (11) into (8), the pdf ofthe channel access delay can be obtained. Then themgf of the total service time is expressed as follows
MR(t) = MY (t)(1− t(Bw
2
β2µ2
)−1)−1.
Similar to the non-aggregated scenario, the expectedservice time of a request in CBOSC queues is
E[DCBOSC ] =λ2
ME[(Tw)2]
2(1− ρ)+ E[Tw].
For DSC queues, the expectation and the secondmoment of the service time are
E[T v] =β2µ2
Bv2Psucc
and
E[(T v)2] =∞∑v
∞∑u
Bv2
β2µ2
e− Bv
2β2µ2
vPsucc
(1− Psucc)u−1(uv)2.
The expectation of the system delay of the DSCqueues is
E[DDSC ] =λ2
N2E[(T v)2]
2(1− λ2
N2E[T v])
+ E[T v].
Similar to the approximation for the aggregatedscenario in heterogeneous RAOSC-DSC networks,the average system delay of the heterogeneous
CBOSC-DSC networks based on the aggregatedscenario is approximated by
Dagg =
{E[DCBOSC ], if E[DCBOSC ] ≥ E[DDSC ],E[DDSC ], otherwise.
TABLE II: Values of the parameters used in thesimulation
Room size 8×10×2.5 metersNumber of CBOSC APs 10
CBOSC bandwidth 20 MbpsMinimum contention window 16Maximum contention window 1024
RTS size 44 bytesCTS size 38 bytes
DIFS 50 µsecSlot size 20 µsec
Number of DSC APs 20Reuse factor of DSC 4Modulation scheme 4-PAM
Maximum optical power 0.5 WattNoise level 4.7×10−14 A2
Field of view 40 degreesDSC bandwidth 10 MbpsPsucc of DSC 0.5
request arrival rate 0.05/slotmean request size 1000 bytes
C. Empirical AnalysisTo validate our analytical model and compare
the system delay performance of heterogeneousCBOSC-DSC networks under non-aggregated andaggregated scenarios, we conduct extensive simu-lations under the homogeneous traffic assumptions.The final system delay is averaged over 100,000simulated requests. For the simulation settings, weconsider a 8 × 10 meters room. There are 10CBOSC APs located in a single contention domain(i.e. each pair of CBOSC APs have non-negligibleinterference between each other). For 802.11 a/g/n,the minimum and maximum contention windowsizes [38] are 16 slots and 1024 slots, respectively.Referring to [20], the 802.11 MAC settings, in-cluding RTS size, CTS size, DIFS and slot size,are set to 44 bytes, 38 bytes, 50 µsec and 20µsec, respectively. In the room, there are 20 DSCAPs mounted on the 2.5 meters height ceiling ingrid structure, where each DSC AP is serving a2 × 2 meters square area. Each adjacent 4 DSCAPs are using different frequency. In other words,the reuse factor is 4. Each DSC AP has 5 MHzbandwidth and is using 4-PAM as the modulationscheme. The maximum optical power of each DSC
14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.750
60
70
80
90
Traffic allocation ratio α2
Ave
rage
sys
tem
del
ay (
slot
)
simulationanalytical
(a) non-aggregated scenario
0 0.1 0.2 0.3 0.4 0.5 0.6 0.740
60
80
100
120
140
160
180
Request splitting ratio β2
Ave
rage
sys
tem
del
ay (
slot
)
simulationanalytical
(b) aggregated scenario
Fig. 13: Comparison between the simulation and analytical results of the average system delays for (a)non-aggregated scenario; (b) aggregated scenario
AP is set to 0.5 Watt. The Gaussian noise value iscalculated based on the parameters in [39] and isset to 4.7×10−14 A2. The semi-angle at half power,area of detector, optical filter gain and refractiveindex are all set to the same as the parameter in[39]. For 4-PAM, the required minimum SNR valuefor achieving 10−3 bit error rate is 19.80 dB [40].Based on the setting, the SNR value for the userterminals located at the boundary of each AP’scoverage is 25.78 dB, which satisfies the minimumrequirement of 4-PAM. The field of view (FOV) ofoptical receivers is set to 40 degrees. which meansthat for each DSC AP, the signals from the closestinterfering AP will not be received by the servinguser terminals. Therefore, each DSC AP can achieve10 Mbps throughput. Within each 2 × 2 meterssquare area served by each DSC AP, based on thepractical settings given above, the data rate of auser terminal will be the same no matter where itis located. The uniformly distributed blockage rateis set to 0.5. All the parameter settings for CBOSCand DSC networks are given in Table II.
In Fig. 13, we vary the traffic allocation ratioα2 for the non-aggregated scenario and the requestsplitting ratio β2 for the aggregated scenario, andcompare the simulation and analytical results forthe average system delay. For both scenarios, wecan see the close match between the analytical andsimulation results. The simulation results are theaverage system delay over all the simulated requests.
If the number of simulated requests is large enough,the simulation results are expected to converge to theanalytical results. Refer to (9) in [20] as follows,
1
µ=ρ(N − 1)[TS + TC
p
2(1− p)] + W + TS
+ TCp
2(1− p)
the factor of 2 in the denominator of TCp
2(1−p)
represents the first degree approximation that onlytwo nodes are involved in a collision. The firstdegree approximation underestimates the collisioneffect, thus under some cases (i.e. three or morenodes collide), the simulation result is expected tobe above the analytical one. On the other hand, referto (6) in [20] as follows,
p = 1− P [SE]N−1
where P [SE] denotes the probability that a nodedoes not transmit in a slot, the assumption behind(6) in [20] is that the event that a node does nottransmit in a slot is independent of similar decisionsby the other nodes. The decoupling approxima-tion overestimates the collision probability, thereforeunder some cases (i.e. a node does not transmitis correlated to the similar decisions of the othernodes), the simulation result is expected to be belowthe analytical one. As expected, there exist optimalvalues of α2 and β2 that will lead to the minimum
15
0.04 0.06 0.08 0.1 0.12 0.14 0.1630
40
50
60
70
80
90
Total arrival rate λ2
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon−aggregation
(a) λ2
500 1000 1500 200020
40
60
80
100
120
Mean request size µ2 (byte)
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon−aggregation
(b) µ2
0 5 10 15 20
CBOSC bandwidth Bw2
(Mbps)
35
40
45
50
55
60
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon-aggregation
(c) Bw2
2 4 6 8 10
DSC bandwidth Bv2(Mbps)
0
50
100
150
200
250
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon-aggregation
(d) Bv2
Fig. 14: Comparison between the average system delays of non-aggregated scenario and aggregated scenarioin terms of (a) λ2; (b) µ2; (c) Bw
2 ; (d) Bv2 , when M = 10 and N2 = 20
average system delay of the heterogeneous CBOSC-DSC network. With α2 and β2 lower than the opti-mal values, the DSC network will contribute moredelay penalty to the average system delay. However,since the contention and backoff mechanism is notutilized in DSC, the average system delay will notapproach to infinity even if α2 and β2 are equal to 0.In contrast, as α2 and β2 increase above the optimalvalue, the CBOSC queues will be saturated quickly,which leads to infinite average system delay.
In Fig. 14, the values of λ2, µ2, Bw2 , B
v2 are ini-
tially set to the values in Table II. In each plot,one of these four parameters is varied while keepingthe other three fixed at the initial values. In Fig. 14
(a), it is observed that the average system delay ofaggregated scenario is not always lower than thatof the non-aggregated scenario. This is the majordifference from the simulation results of heteroge-neous RAOSC-DSC networks, where contention andbackoff mechanism is not utilized. As the requestarrival rate increases, the backoff penalty broughtby aggregation will surpass the benefit from splittingthe requests. Therefore, in heterogeneous networkswhere contention and backoff mechanism is applied,under certain conditions, the non-aggregated sce-nario outperforms the aggregated scenario in termsof average system delay. In Fig. 14 (b), as the meanrequest size increases, the gap between aggregation
16
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080
50
100
150
200
Total arrival rate λ2
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon−aggregation
(a) λ2
400 600 800 1000 1200 1400 1600 18000
50
100
150
200
250
Mean request size µ2 (byte)
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon−aggregation
(b) µ2
0 5 10 15 20
CBOSC bandwidth Bw2
(Mbps)
40
60
80
100
120
140
160
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon-aggregation
(c) Bw2
10 12 14 16 18 20
DSC bandwidth Bv2 (Mbps)
30
40
50
60
70
80
90
Min
imum
sys
tem
del
ay (
slot
)
aggregationnon-aggregation
(d) Bv2
Fig. 15: Comparison between the average system delays of non-aggregated scenario and aggregated scenarioin terms of (a) λ2; (b) µ2; (c) Bw
2 ; (d) Bv2 , when M = 2 and N2 = 4
and non-aggregation increases. These results areopposite to the results of Fig. 8 (b). The reason isthat as the mean request size decreases, the benefitbrought from aggregation becomes less evident thanthe backoff penalty. In Fig. 14 (c) and Fig. 14 (d),the results are consistent with the results of Fig. 8(c) and (d). As the CBOSC bandwidth increases,the collision probability of the CBOSC networkdecreases. Thus, the delay penalty effect brought byaggregation is diminishes. As the DSC bandwidthincreases, similar to the heterogeneous RAOSC-DSC network, the benefit gain of aggregated sce-nario is slightly reduced. This is because the increasein the DSC bandwidth leads to smaller optimal α2
and β2, which will reduce the gap between thedelay performance of non-aggregated scenario andaggregated scenario.
To evaluate the effect of the number of APs onthe system delay performance of the heterogeneousCBOSC-DSC network, we reduce the number ofCBOSC APs M from 10 to 2 and the number ofDSC APs N2 from 20 to 4. The comparisons be-tween non-aggregated scenario and aggregated sce-nario in terms of λ2, µ2, B
w2 , B
v2 are performed again
and the simulation results are shown in Fig. 15.Compared to the simulation results when M = 10and N2 = 20, the average system delays are higherwhen M = 2 and N2 = 4. This is because the
17
3 4 5 6 7 8 9 10Number of OSCC APs M
39
40
41
42
43
44
45
46
47M
inim
um s
yste
m d
elay
(sl
ot)
aggregationnon-aggregation
(a) M
10 12 14 16 18 20Number of DSCB APs N
2
38
40
42
44
46
48
50
52
Min
imum
sys
tem
del
ay (
slot
) aggregationnon-aggregation
(b) N2
Fig. 16: Comparison between the minimum average system delays of non-aggregated scenario andaggregated scenario in terms of (a) the number of CBOSC APs M ; (b) the number of DSC APs N2
total network capacity is reduced when the num-ber of APs decreases. We also observe that whenM = 10 and N2 = 20, the benefit gain of aggregatedscenario over non-aggregated scenario is less than20%; while this benefit gain increases up to 40%when M = 2 and N2 = 4. In addition, we set thevalues of λ2, µ2, B
w2 , B
v2 to 0.05/slot, 1000 bytes,
20 Mbps, 10 Mbps, respectively. The number ofCBOSC APs M are varied from 3 to 10 while fixingthe number of DSC APs N2 to 20. The simulationresults are shown in Fig. 16 (a). As it is expected,the gap between aggregation and non-aggregationis decreasing when the number of CBOSC APs Mincreases. This is because with certain value of totalrequest arrival rate, mean request size, CBOSC andDSC bandwidth, the collision probability of CBOSCnetwork is increasing as the number of CBOSCAPs increases. In particular, the backoff penalty ofaggregated scenario is dominating as the number ofCBOSC APs increases. Therefore, the benefit gainof aggregated scenario over non-aggregated scenariobecomes dominant when the number of CBOSCAPs is small. In Fig. 16 (b), the number of DSC APsN2 are varied from 10 to 20 while fixing the numberof CBOSC APs M to 10. We observe that thegap between aggregation and non-aggregation doesnot change much when the number of DSC APsN2 varies. However, the minimum average systemdelay of the two scenarios are both decreasing as
N2 increases. This is due to the additional networkcapacity added by increasing number of DSC APs.
Furthermore, to evaluate the effect of other dis-tribution of arrival process on our approach, weinvestigate two other distributions of interarrivaltime by simulations - generalized pareto distribution[41] and weibull distribution [42]. The pdf of thegeneralized pareto distribution is as follows:
ypareto =f(x|k, λpareto, θ)
=λpareto(1 + k(x− θ)λpareto)−1− 1
k
where k is the shape parameter, λpareto is the recip-rocal of the scale parameter and θ is the thresholdparameter.
The pdf of the weibull distribution is shown asfollows:
yweibull =f(x|λweibull, b)
=λweibullb(λweibullx)b−1e−(λweibullx)
b
where λweibull is the reciprocal of the scale parameterand b is the shape parameter.
In the simulation, under the assumption of gen-eralized pareto distribution of interarrival time, weset k = 1 and θ = λk. Under the assumption ofweibull distribution of interarrival time, we set b= 1.5. Similar to the evaluation performed above,the minimum average system delay performance of
18
0.1 0.2 0.3 0.4 0.5λ
Pareto(per sec)
2
4
6
8
10
Num
ber
of V
LC
hot
spot
s
0.62
0.64
0.66
0.68
0.7
The
rat
io o
f ag
g to
non
-agg
(a) Generalized Pareto distribution
0.1 0.2 0.3 0.4 0.5λ
Weibull(per sec)
2
4
6
8
10
Num
ber
of V
LC
hot
spot
s
0.55
0.6
0.65
0.7
0.75
0.8
The
rat
io o
f ag
g to
non
-agg
(b) Weibull distribution
Fig. 17: For the case of RAOSC-DSC, the ratio of the minimum average system delay of the aggregatedscenario to that of the non-aggregated scenario in terms of (a) λpareto for generalized pareto distributionand (b) λweibull for weibull distribution, with N1 varied from 1 to 10
0.05 0.1 0.15λ
Pareto
30
35
40
45
50
55
60
Min
imum
sys
tem
del
ay(s
lot) aggregation
non-aggregation
(a) Generalized Pareto distribution
0 0.02 0.04 0.06 0.08 0.1 0.12λ
Weibull
20
30
40
50
60
70
Min
imum
sys
tem
del
ay(s
lot) aggregation
non-aggregation
(b) Weibull distribution
Fig. 18: For the case of CBOSC-DSC, comparison between the minimum average system delay of non-aggregated scenario and aggregated scenario in terms of (a) λpareto for generalized pareto distribution and(b) λweibull for weibull distribution, when M = 10 and N2 = 20
19
non-aggregated scenario and aggregated scenario isevaluated for the RAOSC-DSC case and CBOSC-DSC case. The other simulation settings are thesame as the settings above. The simulation resultsare shown in Fig. 17 and Fig. 18. It can be observedthat based on the given simulation settings for thecase of RAOSC-DSC the minimum average systemdelay of the aggregated scenario is still always lowerthan that of the non-aggregated scenario, while forthe case of CBOSC-DSC, the minimum averagesystem delay of the aggregated scenario is lowerthan that of the non-aggregated scenario for lighttraffic condition and vise versa. These results areconsistent with the results based on the assumptionof Poisson arrival process.
V. CONCLUSION
In this paper, two cases of heterogeneous OSC-DSC wireless networks are considered for aggre-gation and non-aggregation scenarios. In the firstcase, the heterogeneous RAOSC-DSC network isinvestigated. Given the assumptions that requestsarrive according to Poisson process and the re-quest size is exponentially distributed, it is provedthat the minimum average system delay of theaggregated scenario is always lower than that ofthe non-aggregated scenario. An efficient methodis proposed to approximate the optimal requestssplitting ratio in the aggregated scenario. The an-alytical results when applying the non-aggregatedscenario and simulation results when applying theaggregation system are also presented. In the secondcase, the heterogeneous CBOSC-DSC network isstudied. The average system delay is derived forboth the non-aggregated and aggregated scenarios.Extensive simulation results imply that, when con-tention and backoff mechanism is considered, thenon-aggregated scenario outperforms the aggregatedone under certain conditions. This is because thebackoff penalty caused by aggregation exceeds thebenefit from splitting the request.
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