In this section, we formulate a non-linear programming framework to find the optimized relay selection and flow scheduling which minimize the extension of total transmission time and maximize the system throughput. Our proposed framework is applicable to the muti-piconet communications with directional antennas and time-sharing in a superframe. We first define our object and then develop the constraints. The symbols and notations we used are listed in Table 3.1..
3.1 Framework Objective
As we mentioned in Chapter 1, our algorithm is divided into two phases. In phase 1, we optimally schedule intra-piconet flows and minimized the CTAP. After scheduling all intra-piconet flows, we can derive CTA sets which contain the concurrent transmission intra-piconet flows within a CTA and the CTA duration. We do not know how many CTA sets will be used finally, so we allocate every hop of all inter-piconet flows a virtual CTA of which the duration is initially zero at first. The duration of the virtual CTAs will be assigned after we schedule the flows in. We assume there are t CTAs that t= + and equals to the sum of the inter-piconet flows.
Here we list the assumptions and givens before optimization.
(1) The piconet path of an inter-piconet flow is given but the relay node is undecided.
That is, we only know the flow would pass through which piconets but relay nodes. We also get the hop number of an inter-piconet flow from the piconet path.
(2) The piconet topology, flow requests (including the senders and receivers), DEVs’
type, and relay-capable DEVs are known by the PNC of that piconet. The reception and transmission beam width are set to be the same.
(3) We assume all piconets in the system are synchronized by control messages.
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(4) After scheduling in phase 1, the minimized CTAP, CTA sets and their time durations are given.
(5) We schedule the inter-piconet flows hop by hop, and take the last hop of an inter-piconet flow as an intra-piconet flow.
We seek to minimize the extension of CTAP and the extension can be caused by scheduling the inter-piconet flows into either the existing CTAs with intra-piconet flows or the virtual CTAs and turn them to real. Let be the time of extended by flow i. The object is
Min. ∑ (1)
s.t. 1, 0, , 1, (2)
∑ ∑ , (3)
= ∆ , ∆ , , intra-piconet flow in (4)
= ∆ , ∆ , , intra-piconet flow in (5)
· = 1 (6)
∑ , 1 (7)
= , 0, (8)
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Then, we introduce two classes of constraints that the relay selection and flow scheduling have to satisfy and optimize the object.
z Constraints for Basic Relay Selection
In this part, we list two basic constraints that the feasible solution has to follow, but decisive constraints for the best relay selection are derived in next section.
1. Single path constraint
Although the piconet path of a flow transmission is known and there are several relay candidates can be selected, the multi-path issue is ignored in our research. Only one path is determined by the formulation. We use and to denote the outgoing and incoming flow set of device If the device is a sender instead of a receiver, and are 1 and 0 respectively, and vise versa; if the device is a relay node for more than one flow, then the sum of outgoing flows is equal to the
Table 3.1 Symbols and notations used in formulation Time extension of by flow i
T CTA set, T={ , , …, }, t= +
Number of CTAs for all scheduled intra-piconet flows Number of the inter-piconet flows
Set of outgoing flow number on device Set of incoming flow number on device S Set of senders
Sender of flow i D Set of destinations
destination of flow i R Set of relay candidates
, Flow i’s identifier, flow i selects relay candidate r of the requested hop in .
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sum of incoming flows. The constraint is shown in (2).
2. Path guaranteed constraint
We have to guarantee all inter-piconet flows are all well scheduled with adequate relay nodes. For flow i, the is an identifier that means a flow i taking relay node r transmitting in . The sum of flow identifiers in CTAs and for all flows should equal to the flow requests and the constraint can be expressed as (3).
z Constraints for Optimal Relay Selection and Flow Scheduling 3. Interference-free constraint
Flows can transmit data simultaneously if two flows do not interfere to each other. Here we define an interference-free condition; two flows can be scheduled in the same CTA if they satisfy the condition. A receiver only accepts signals in a certain beam direction, because it equips a directional antenna. We assume the reception and transmission beam widths are .
Let ( , ) and ( , ) be the senders and receivers of flow and . The interference-free condition is twofold:
(1) If (or ) is not located in ’s (or ’s) reception beam, (or ) cannot hear the signal from (or ) and two flows do not interfere with each other.
(2) If (1) is not satisfied, but (or ) is not located in s (or s) sending beam, then and still do not interference. Otherwise, and can not be scheduled in the same CTA.
The inter-piconet flow scheduled in a CTA has to verify the interference-free condition with all intra-piconet flow within the CTA. We use the symbol ∆ , to represent the angle between two vectors and , and according to the trigonometric function, ∆ , cos | || |· . For flows and , we can derive
four angles, ∆ , , ∆ , , ∆ , , and ∆ , . In (4), if ’s transmission beam
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does not cover or does not located in ’s reception beam, then is set to be 1. On the other hand, if and follow the condition, is set to be 1 in (5).
Constraint (6) shows that flow i takes relay r in if (4) and (5) are satisfied.
4. Half-duplex constraint
We assume every DEV direct to a beam sector only, and apply the half-duplex constrain of the wireless communications to 60GHz WPAN. That is, more than two flows share the same DEV can not be scheduled in a CTA, which also means that a DEV can either be a sender or receiver of one flow in the same time duration. We derived the constraint as (7), where denotes the scheduled flow in .
5. CTAP extended by flow i
This constraint couples CTAP extension with scheduling result and this is also a key constraint that decides which relay and transmission timing combination. If an inter-piconet flow with relay r is allocated in , it may increase the time and also affect the accommodation to other flows. The CTAP extension is correlated to the CTA extension and new CTA allocation by scheduling each inter-piconet flow. In (8), if flow i is scheduled and the transmission time is larger than the CTA duration, the extension time is the differences of the two.
z Complexity
If there are m relay candidates, n inter-piconet flows, and t CTAs for intra-piconet flows, the complexity of finding the optimal solution is O( ! ). The computation includes ! scheduling orders, relay selection combination, and each flows has ( ) CTAs can be scheduled in. Obviously, the optimization is a NP problem, and the computation in a PNC would cause large power consumption.
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