1
Impact of Information Availability on StarvationMitigation and Delay Minimal Delivery in ICRCNs
Ribal Atallah and Wissam Fawaz
AbstractβThis paper looks into an Intermittently ConnectedRoadside Communication Network (ICRCN) scenario comprisingtwo isolated source Stationary Roadside Units (SRUs) relying onmobile smart vehicles to relay data to a destination SRU. In thiscase, it was shown in [1] that the downstream source SRU maysuffer from a significant starvation problem. As such, a Markovdecision process framework was established therein to identify asuitable Bulk Release Decision Policy (BRDP). BRDP was thenimplemented within a Starvation Mitigation and Delay-Minimal(SMDM) delivery scheme.
In this paper, we investigate the impact of the level ofinformation availability at the upstream non-starving node onthe performance of the SMDM scheme. In particular, extensivesimulations are conducted for the purpose of quantifying theability of SMDM to jointly mitigate starvation and achieveminimal end-to-end bundle delivery delay under conditions ofperfect, imperfect, and no information availability at the non-starving node.
Index TermsβICRCN, SRU, Performance Evaluation, MDP.
I. INTRODUCTION
THE conception of ICRCNs consists of transforming mobilevehicles into smart and communication-enabled entities
that are able to establish connectivity among isolated Station-ary Roadside Units (SRUs). ICRCNs emerged as such as aterrestrial application of the Disruption-Tolerant Networkingparadigm, [2]. Precisely, this conception consists of augment-ing vehicles, particularly those restricted to navigable road-ways, with computerized modules, finite buffers and wirelesscommunication devices to inter-communicate as well as tocommunicate with the SRUs deployed along the roadways.Under such conditions, ICRCNs represent an effective andcost-minimal solution for bringing digital connectivity closerto isolated areas, especially when the cost of setting up sucha networking infrastructure is deemed elevated [3].
This paper revolves around the networking scenario de-picted in Figure 1, which involves three SRUs, namely twosource SRUs referred to as π1 and π2 along with a destinationSRU, π·. All three SRUs are installed along a one-dimensional,uninterrupted highway segment. The three SRUs cannot di-rectly communicate with one another as each one is locatedoutside the respective coverage ranges of the two others. Thetwo source SRUs π1 and π2 are assumed to be completelyisolated, while π· is privileged by a connection to the Internet.In the absence of a networking infrastructure connecting π1
and π2 to π·, mobile vehicles act as store-carry-forward datacarriers from both of π1 and π2 to π·.
R. Atallah and W. Fawaz are with the ECE department of the LebaneseAmerican University. E-Mail: [email protected], [email protected]
978-1-4799-3060-9/14/$31.00 Β©2014 IEEE
D
S2
S1
d2
d1
WirelessAccess Point
SRU
CityMetropolitanArea Network
THE INTERNET
Fig. 1. ICRCN scenario.
Upon the occurrence of a vehicle arrival to either one ofthe two source SRUs π1 or π2, the vehicle presents itself tothe SRU as a bundle1 release opportunity. In a point-to-pointICRCN with a single source SRU, the work of [3] assessedthe contribution of such an opportunity to the minimizationof the average end-to-end delivery delay of a singly releasedbundle. The authors of [4] addressed the limitations of [3]and demonstrated how the release of a bulk of bundles2 froma source SRUβs buffer significantly improves the performancein terms of the average end-to-end delivery delay.
While these studies focused only on ICRCN scenariosconsisting of a single source SRU, the work of [1] optedfor the investigation of an ICRCN scenario where severalsource SRUs may be present along the way to the destinationSRU. In addition, unlike the previous studies, the authors in[1] considered the case in which vehicle buffers are finiteand thus unable to accommodate all the possibly emanatingbundles from each one of the encountered source SRUs. Undersuch circumstances, it becomes highly likely that the sourceSRUs that are located further downstream be subject to severestarvation especially if the upstream SRU starts downloadingtoo much data to passing by vehicles and thereby hogs thefinite buffers offered by these vehicles.
More precisely, considering the highway segment illustratedin Figure 1, vehicles navigating towards π· will first pass byπ1 before encountering π2. Both of π1 and π2 will try to loadthe arriving vehiclesβ finite buffers with as many as possibleof their respective data bundles. Such a bundle release model
1Data and control signals are combined in a single atomic entity, calledbundle, that is transmitted across a DTN-based ICRCN, [2].
2A group of bundles is referred to as a bulk of bundles.
2
eventually favors π1 over π2. This is particularly true sinceby the time a vehicle has reached π2, its buffer might havebeen considerably loaded (ultimately fully exhausted) withπ1βs bundles. A node such as π2 is typically called a starvingnode, since bundles would rapidly accumulate in π2βs buffer.As a result, incoming bundles to π2 will be subject to excessivequeueing delays.
The study in [1] proposed to mitigate the severity of thestarvation that a downstream SRU may suffer from throughthe deployment of the so-called Starvation Mitigation andDelay-Minimal (SMDM) delivery scheme. In the context ofSMDM, the authors assumed that π1 possesses real-time fullknowledge (i.e., perfect information) about the number ofbundles queueing at π2
3. Through the availability of perfectinformation at an upstream node, SMDM was shown to beable to resolve the starvation problem while at the same timeachieving minimal end-to-end bundle delivery delay.
We refer in the sequel to the variant of SMDM whereπ1 is equipped with perfect information about π2βs queuelength as SMDM-PI. In practice, it is impossible for π1 toacquire such perfect information in a real-time manner. Thislimits the SMDM-PI scheme in terms of practicality andimprisons its ability to achieve optimal performance resultswithin the frames of a purely theoretical study. Inspired by thisobservation, the present paper investigates a new variant of theSMDM scheme dubbed SMDM-II that differs from SMDM-PIin that imperfect information is assumed to be present at π1
with respect to the status of π2βs buffer. A simulation studyis laid out in order to evaluate the impact of the existence ofimperfect information at π1 on the performance of the SMDMscheme.
The rest of this manuscript is organized as follows. Insection II, the SMDM scheme proposed in [1] is described.Section III introduces the SMDM variant proposed in thispaper. In section IV, extensive simulations are performed toanalyze the performance of the bundle delivery schemes understudy. Finally, section V concludes the paper.
II. STARVATION MITIGATION DELAY MINIMAL BUNDLE
DELIVERY
A. Background
In the context of the ICRCN scenario illustrated in Figure1, π1 and π2 are considered to be identical. Their respec-tive coverage ranges are of length ππΆ(πππ‘πππ ) each. Thedistances that separate π1 and π2 from π· are π1(πππ‘πππ )and π2(πππ‘πππ ) respectively. Following the guidelines givenin [1], [3], [4], the following assumptions are made:
β A1: The bundle arrivals follow a Poisson process with arate of ππ
(ππ’πππππ π πππππ
).
β A2: π1 and π2 are subject to equal bundle arrival rates.β A3: The bundle sizes are fixed to π (ππ¦π‘ππ ).β A4: π1 and π2 have equal data transmission rates of
ππ
(ππππ‘π π πππππ
).
β A5: π1 and π2 are equipped with infinite buffers.
3Thereafter, the number of bundles stored in π2βs queue will be referredto as π2βs queue length.
β A6: The vehicle arrivals follow a Poisson process with arate of ππ£
(π£πβπππππ π πππππ
).
β A7: The per-vehicle speed ππ is drawn froma uniform distribution defined over the range[ππππ;ππππ₯]
(πππ‘πππ π πππππ
).
β A8: The per-vehicle capacity πΆπ is drawn from a uniformdistribution defined over the range [πΆπππ;πΆπππ₯].
β A9: Each vehicle maintains a constant speed during itsnavigation over the highway segment [π1π·].
β A10: The IEEE 802.11π protocol is used for vehicle-to-SRU communication.
B. Theoretical foundation
On the occurrence of a vehicle arrival, either one of π1
or π2 has to decide on whether or not to use the arrivingvehicle as a bulk transporter to the destination SRU π·. Thisrelease decision process evolves over the sequence of vehicles{0, 1, 2, ..., π, ...} with speeds {π£0, π£1, π£2, ..., π£π, ...} arrivingat times {π‘0, π‘1, π‘2, ..., π‘π, ...}. Based on assumption (A6),{πΌπ, π β₯ 1}, the inter-arrival time intervals, form a sequenceof independent and identical exponentially distributed valueswith an average πΈ[πΌ] = 1
ππ£. Let ππ,π denote the number of
bundles waiting in the ππ‘β source SRUβs buffer (π = 1, 2) attime π‘π. Also let ππ,π represent the decision made by the ππ‘β
SRU (π = 1, 2) at time π‘π and ππ,π represent the size of thebulk released by the ππ‘β source SRU at that time. In the contextof the ICRCN scenario given in Figure 1, ππ,π β {0; 1}. It isimportant to highlight in this regard that whenever ππ,π = 0,ππ,π = 0 is the only possible decision to be made and thus nobundles are released. As such:
ππ,π+1 = πππ₯{0, ππ,π βππ,π β ππ,π}+π΄π,π (1)
where π΄π,π is the number of newly incoming bundles duringthe time interval πΌπ+1 = π‘π+1 β π‘π. π΄π,π has an average ofπ΄π = πΈ[π΄π,π] = πππΈ[πΌ]. In view of the above, the state ofthe middle source SRU π2 can be described by the two-tuple(π2,π;ππ) and its dynamics are governed by equation (1).
Now, define:
πΆ2 (οΏ½ΜοΏ½π, π£π, π¦π) =οΏ½ΜοΏ½π
π΄2
+ π2(π£π, π¦π) (2)
πΆ2 (οΏ½ΜοΏ½π, π£π, π¦π) is the single stage cost function correspondingto π2βs state (π2,π = οΏ½ΜοΏ½π;ππ = π£π) and release decision π2,π =π¦π. It is composed of two terms, namely: a) οΏ½ΜοΏ½π
π΄2being the
cost associated with π2βs queue length (or the queueing delay)and b) π2(π£π, π¦π) corresponding to the cost incurred by theexploitation of the vehicle of speed π£π as a bundle carrier fromπ2 to π· (i.e., the transit delay). Note that π2(π£π, π¦π) =
π2
π΄2β π¦π
π£π.
At this level, given that ππ is uniformly distributed overa finite range, the vehicle inter-arrival time is exponentiallydistributed, and the decision space π2,π is finite, there existsa policy πβ
2 that minimizes the following cost function at π2
[3], [5]:
πΆ2 =1
π΄2
lim supπββ
[1
πβ πΈ
[πβ
π=0
(π2,π +
π2ππ
β π2,π
)]](3)
3
The problem of identifying the policy πβ2 that minimizes (3)
is a classical average cost Markov decision problem where ateach time step, π2 chooses an action, namely, a release/no-release decision, and incurs a per-stage cost similar to theone given by (2). Such problems can be solved via dynamicprogramming techniques like the ones discussed in [6].
Similarly to π2, a single stage cost function can be for-mulated for π1. However, in addition to π1βs queue sizeand release decision, this cost function has to account forπ2βs queue size and release decision as well. It is, therefore,expressed as:
πΆ1(οΏ½ΜοΏ½π, οΏ½ΜοΏ½π, π£π, οΏ½ΜοΏ½π, π¦π) =οΏ½ΜοΏ½π
π΄1
+π1(π£π, οΏ½ΜοΏ½π)+πΆ2 (οΏ½ΜοΏ½π, π£π, π¦π) (4)
Similar to (2), the term οΏ½ΜοΏ½π
π΄1in (4) is tied to the queue length
of π1. Also, in the same spirit, π1(π£π, οΏ½ΜοΏ½π) =π1
π΄1β οΏ½ΜοΏ½π
π£π. Follow-
ing similar arguments as above, there exists a deterministicMarkov policy πβ
1 β‘ πβ1(οΏ½ΜοΏ½π, οΏ½ΜοΏ½π, π£π, οΏ½ΜοΏ½π, π¦π) that minimizes the
average cost function developed for π1 and which is given by:
πΆ1 =1
π΄1
lim supπββ
[1
ππΈ
[πβ
π=0
π1,π +π1ππ
β π1,π
]]+ πΆ2 (5)
In this way, the joint starvation mitigation/delay minimizationproblem at π1 is formulated as a Markov Decision Processframework with an average cost criterion given by (5). Follow-ing the guidelines of [6], this problem can be resolved using aDynamic Programming approach. The Bulk Release DecisionPolicy (BRDP) for the overall system of the two source SRUsis represented by πβ β‘ (πβ
1 , πβ2). Next, BRDP is implemented
within a Starvation Mitigation Delay-Minimal (SMDM) bulkbundle delivery scheme.
In the context of the SMDM delivery scheme, the bundlerelease decisions made by both source SRUs π1 and π2 aredriven mainly by πβ. If a decision is made at time π‘π to releasea bulk of bundles to an arbitrary vehicle π with a free buffercapacity πΆπ, the number of bundles ππ,π to be downloaded tothat vehicle by the ππ‘β SRU (π = 1, 2) is determined as follows.At π1, π1,π basically depends on: a) πΆπ being the amount offree buffer space that is available at the πth vehicle as well ason b) π2,π being the number of bundles currently queueing atπ2. More specifically, the number of bundles released by π1
to a vehicle π selected by BRDP is governed by the followinginequality:
π1,π β€ π1,π
π1,π +π2,πβ πΆπ (6)
π2,π plays a major role in throttling π1 to which, now, isallocated only a well defined portion of the free buffer spaceavailable at the vehicle chosen by BRDP. This, by itself,contributes to the reservation of some spare buffer space forπ2. Consequently, π2 will, in turn, beneficially exploit themore abundant remaining vehicle buffer capacity spared byπ1 and hence abandon the title of a starving node.
C. Inadequacy of Perfect Information Assumption
It is clear that as described above, SMDM was builtaround the unrealistic assumption that the upstream SRU π1
is equipped with accurate instantaneous information about thequeue length at the downstream node π2. This assumption ofperfect information in turn allowed SMDM to realize effec-tive starvation mitigation. Obviously, a more realistic SMDMvariant would be one that is based on imperfect information.Nonetheless, the more information is made available at π1
about π2βs buffer status, the higher the accuracy of π1βsknowledge will be. It is thus indispensable for any SMDMvariant based on imperfect information to identify practicalmeans for feeding π1 with as much information as possibleabout the status of π2βs buffer. This observation establishedthe foundation for the development of a new SMDM variantunder which π1 would only have partial imperfect informationand yet be able to contribute to a great extent to the mitigationof the starvation problem. The aforementioned SMDM variantthat we refer to as SMDM-II is proposed next.
III. SMDM WITH IMPERFECT INFORMATION
Recall that the objective of this study is to introduce a newvariant of the SMDM delivery scheme that operates underthe premise of the availability of imperfect information atπ1 with respect to the status of π2βs buffer. The majorityof todayβs highways are characterized by the use of dualcarriageways running in opposite directions for the purpose ofaccommodating traffic plying between any two geographicalpoints. As a result, it becomes possible to create a feedbackchannel connecting π2 to π1 by taking advantage of thevehicles navigating from π2 to π1 and by having these vehiclestransport non-real time information about π2βs buffer status toπ1.
More formally, under the proposed SMDM-II, π1 wouldmaintain a variable called πΈπ π‘ππππ‘πππ2 that represents thebest current estimate by π1 of π2βs buffer length. When a newvehicle carrying a new πππππππ2
4 from π2 arrives at π1, π1
simply updates πΈπ π‘ππππ‘πππ2 as follows: πΈπ π‘ππππ‘πππ2 =πππππππ2. In addition to having an estimate of π2βs bufferlength, π1 is also required to keep track of the time instantat which πΈπ π‘ππππ‘πππ2 was last updated. As such, each timeπΈπ π‘ππππ‘πππ2 gets updated, π1 sets a variable called π‘π’ππππ‘πto be equal to the time instant at which the update took place.
On the other hand, πΈπ π‘ππππ‘πππ2 is updated differentlywhen a vehicle navigating from π1 to π2 gets to π1. Particu-larly, when a vehicle π navigating towards π2 enters the com-munication range of π1 at time π‘π, π1 updates πΈπ π‘ππππ‘πππ2
according to the following formula:
πΈπ π‘ππππ‘πππ2 = πΈπ π‘ππππ‘πππ2 + π2 β (π‘π β π‘π’ππππ‘π) (7)
where π2 β (π‘π β π‘π’ππππ‘π) represents the average number ofbundles that accumulates in π2βs buffer over the time interval[π‘π’ππππ‘π; π‘π].
So, with SMDM-II, π1βs imperfect information about thequeue size at π2 takes the form of πΈπ π‘ππππ‘πππ2. Hence,
4The length of the queue at π2 as seen by the newly arriving vehicle.
4
when applying the SMDM strategy as described in sectionII, π1 would employ πΈπ π‘ππππ‘πππ2 instead of the π2,π
parameter introduced earlier wherever the latter appears in theformulation of SMDM.
IV. SIMULATION AND NUMERICAL ANALYSIS
An ad-hoc Java-based in-house discrete event simulator isdeveloped to simulate the scenario illustrated in Figure 1. Themain purpose is to evaluate the performance of the proposedSMDM-II scheme and compare it to that exhibited by anumber of benchmark delivery schemes including the SMDM-PI strategy.
A. Simulation Parameters
The following values were borrowed from [1] for theinput simulation parameters: a) ππ = 30 (bundles/s), b)π = 1500 (bytes), c) ππ = 1 (Mbits/s), d) ππ£ β[0.004; 0.033] (vehicles/s), e) ππππ = 10 (m/s) f ) ππππ₯ =50 (m/s), g) πΆπππ = 0, h) πΆπππ₯ = 30000, i) ππΆ = 200 (m),j) π1 = 2000 (m) and k) π2 = 1000 (m).
It is also important to note that π, the overall load offeredto the system made up of both π1 and π2, is:
π =(π1 + π2)
ππ£πΆ=
2ππ
ππ£πΆ(8)
where ππ is the bundle arrival rate to the ππ‘β SRU (π = 1, 2)and πΆ is the average vehicle spare storage capacity. It is finallyimportant to highlight that the values of the input parameterswere set to ensure that the resulting system of SRUs is stablewith 0 < π β€ 1.
B. Metrics and Benchmark Schemes
The following metrics are used as performance indicators:β Average bundle queueing delay, denoted by ππ·: Average
amount of time a bundle spends in either one of thequeues belonging to π1 or π2.
β Average bundle transit delay, denoted by ππ·: Averageamount of time required by a bundle to travel from eitherone of π1 or π2 to the destination π·.
β Average bundle end-to-end delay, denoted by πΈπ· =ππ· + ππ·
In each simulation run, 107 bundle arrivals per SRU areconsidered and each plotted value of the above-enumeratedmetrics is the average of several independent simulation runsto secure a very narrow 95% confidence interval.
The following bundle delivery schemes are used as bench-marks:
β SMDM with Perfect Information (SMDM-PI): under thisscheme, π1 is assumed to have perfect information aboutthe status of π2βs buffer length. Moreover, bulk of bundlesare released to those vehicles that contribute the most toachieving the dual purpose of starvation mitigation anddelay-minimal bundle delivery.
β The Greedy Bundle Release Scheme with Bulk BundleRelease (GBRS-BBR) proposed in [4]: under this strat-egy, π1 and π2 take identical actions by releasing bulkof bundles to every arriving vehicle regardless of thevehicleβs speed.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2x 10
4
Ο
ED[Sec]
GBRSβBBR β S1
GBRSβBBR β S2
SMDMβPI β S1
SMDMβPI β S2
SMDMβII β S1
SMDMβII β S2
Fig. 2. Average end-to-end delay under the GBRS-BBR, SMDM-PI, andSMDM-II schemes.
C. Simulation Results
Figure 2 concurrently plots as a function of π the end-to-end delays at π1 and π2 as achieved by all of the GBRS-BBR, SMDM-PI, and SMDM-II schemes. This figure is a clearevidence of the amazing ability of the SMDM-PI and SMDM-II delivery schemes to resolve the starvation problem that themiddle SRU π2 suffers from under the GBRS-BBR scheme.
Indeed, when the GBRS-BBR scheme is deployed, the bun-dles of π2 suffer from excessive end-to-end delays as clearlyindicated by Figure 2. This phenomenon can be demystifiedby closely examining the main components that make up theend-to-end delay, namely the average bundle queueing delaydenoted by ππ· and the average bundle transit delay denotedby ππ·. As a matter of fact, according to Figure 3, the bundlesreleased by π2 under GBRS-BBR are expected to experiencelower transit delays as compared to their π1βs counterpart.This can be justified by the fact that the bundles of π2 travela shorter distance to the destination π· as opposed to the ones
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140
60
80
100
120
140
160
180
200
Ο
TD
GBRSβBBR β S1
GBRSβBBR β S2
SMDMβPI β S1
SMDMβPI β S2
SMDMβII β S1
SMDMβII β S2
Fig. 3. Average transit delay under the GBRS-BBR, SMDM-PI, and SMDM-II schemes.
5
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2x 10
4
Ο
QD
GBRSβBBR β S1
GBRSβBBR β S2
SMDMβPI β S1
SMDMβPI β S2
SMDMβII β S1
SMDMβII β S2
Fig. 4. Average queueing delay under the GBRS-BBR, SMDM-PI, andSMDM-II schemes.
generated by π1. However, Figure 4 reveals that the queueingdelay experienced by the bundles released by π2 is extremelyhigh relative to the queueing delay of the bundles at π1.Knowing that both π1 and π2 are subject to the same bundlearrival rate, the only reason why bundles would accumulateat π2 faster than they do at π1 would be that π2 is unable toclear out bundles as fast as π1 is doing so. This is a directimplication from Littleβs Theorem, [7]. The fact that on theaverage, the queue at π2 is bigger than the one at π1 meansthat π1 is monopolizing the capacity of the arriving vehiclesand hence, leaving little spare room for the bundles queueingat π2.
The ability of the SMDM schemes to resolve the starvationproblem incurred by GBRS-BBR can be explained as follows.SMDM enables π1 to regulate its bundle release rate as afunction of the number of bundles it believes to be queueingat π2. Clearly, under SMDM-PI, π1 is privileged with an exactinformation to this end and as such it can optimally regulateits sending rate with a view to reserving an appropriatelydimensioned part of the arriving vehicleβs storage space tobe filled by π2βs bundles. So, π1 no longer selfishly exhauststhe storage capacity of the arriving vehicles and as a resultenables π2 to abandon the title of a starving node. At thislevel, as far as SMDM-II is concerned, what is relevant to thepresent study is the fact that similar conclusions can be drawnabout the SMDM-II scheme. In other words, under SMDM-II, π1 is instructed to regulate its bundle release rate basedon imperfect information about the length of π2βs buffer. Thisjustifies why the end-to-end delay achieved by SMDM-II isslightly worse than that resulting from SMDM-PI, as clearlysupported by Figure 2. Further insights into the behavior ofSMDM-II can be obtained through a careful examination ofππ· and ππ·, the main components underlying πΈπ·. In fact, it isclear from Figure 3 that the transit delays as achieved by bothsource SRUs π1 and π2 under both the SMDM-PI and SMDM-II schemes are equivalent. This can be simply justified by thefact that both of SMDM-PI and SMDM-II select those vehiclesthat best contribute to the minimization of the average transit
delay to the destination SRU. The difference between the end-to-end delay achieved by SMDM-PI and the one incurred bySMDM-II stems from the sub-optimal behavior that SMDM-IIexhibits with respect to starvation mitigation. This observationis corroborated through the results reported in Figure 4 thatindicate a sub-optimal handling of the starvation problem bySMDM-II. Under this condition, unlike in the case of SMDM-PI, the amount of buffer space spared by π1 under SMDM-IIwill be far from being perfectly dimensioned. In consequence,fewer bundles can be released by π2 and π2βs queue getsdestabilized leading to longer queueing delays. Nonetheless,this increase in terms of queueing delay remains acceptable.
In light of the above, once can confidently argue thatSMDM-II is able to tackle convincingly the dual challengeof starvation mitigation and delay-minimal bundle deliverywhile eliminating the need for the inappropriate assumptionof perfect information availability.
V. CONCLUSION
SMDM is a bundle delivery strategy that was developedfor the purpose of dually reducing the starvation problemand achieving delay-minimal bundle delivery in the contextof ICRCNs. At the heart of SMDM though is the non-pragmatic assumption of perfect information availability at theupstream SRU. In a bid to relax this unrealistic assumption,this paper proposed to equip the upstream source SRU withimperfect information about the buffer status of other inter-posed source SRUs. This gave rise to the so-called SMDM-II (SMDM with Imperfect Information) variant. A simulationstudy revealed the ability of SMDM-II to achieve near-optimalperformance when it comes to starvation resolution as wellas delay-minimal bulk delivery. The development of deliverymechanisms based on imperfect information that are moresophisticated than SMDM-II is possible but the study pre-sented herein is a small humble first step in the right direction.This is especially true since our numerical results proved thepractical realizable SMDM-II to be an appealing alternative tothe theoretical SMDM-PI delivery scheme.
ACKNOWLEDGMENT
The authors would like to thank Dr. Maurice Khabbaz forhis valuable comments on the technical content of the presentpaper and for his help in the setup of Figure 1.
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