Multi-channel Wireless Networks with Infrastructure Support: Capacity and Delay Hong-Ning Dai∗ Raymond Chi-Wing Wong† and Qinglin Zhao∗ ∗ Macau University of Science and Technology, Macau hndai@ieee.org zqlict@hotmail.com † Hong Kong University of Science and Technology, Hong Kong raywong@cse.ust.hk Abstract—In this paper, we propose a novel multi-channel wireless network with infrastructure support, called an MCIS network. To the best of our knowledge, we are the first to study the capacity and the delay of such an MC-IS network. In particular, we derive the upper bounds and the lower bounds on the network capacity of such MC-IS networks contributed by ad hoc communications, where the orders of the upper bounds are the same as the orders of the lower bounds, implying that the bounds are tight. We also found that the capacity of MC-IS networks contributed by ad hoc communications is mainly limited by connectivity requirement, interference requirement, destinationbottleneck requirement and interface-bottleneck requirement. In addition, we also derive the average delay of MC-IS networks contributed by ad hoc communications, which is bounded by the maximum number of hops. I. I NTRODUCTION In this paper, we propose a novel multi-channel network with infrastructure support, which is called an MC-IS network. An MC-IS network consists of common nodes (or nodes), each with a single network interface card (NIC), and infrastructure nodes (or base stations), each with multiple NICs, where infrastructure nodes are connected via a wired network that has much higher bandwidth than a wireless network of common nodes. Both common nodes and infrastructure nodes can operate on different channels. Besides, an MC-IS network has the following additional characteristics. • Each node with a single NIC can communicate with either another node or a base station. But, a node supports only one transmission or one reception at a time. Besides, it cannot simultaneously transmit and receive (i.e., it is in a half-duplex mode). • Each base station with multiple NICs can communicate with more than one node. In addition, a base station can work in a full-duplex mode, i.e., transmissions and receptions can occur in parallel. Take Fig. 1 as an example of MC-IS networks. In this network, n nodes are randomly, uniformly and independently distributed on a unit square plane A. Each node is mounted with a single interface that can switch to one of C available channels and it can be a data source or a destination. All the nodes have the same transmission range. Besides, there are b base stations, where b is assume to be a square of an integer b0 (i.e., b = b20 ). Each base station is equipped with m interfaces, each of which is associated with a single interface Fig. 1. Network topology of an MC-IS network that can operate on one of C channels. The plane A is evenly partitioned into b equal-sized squares, which are called BScells. We assume that a base station is placed at the center of each BS-cell. Unlike a node, a base station is neither a data source nor a destination and it only helps forwarding data for nodes. All the base stations are connected through a wired network without capacity constraint and delay constraint. There are two kinds of communications in an MC-IS network: (i) Ad hoc communications between two nodes, which often proceed in a multi-hop manner; (ii) Infrastructure communications between a node and a base station, which span a single hop, as shown in Fig. 1. An infrastructure communication consists of an uplink infrastructure communication from a node to a base station, and a downlink infrastructure communication from a base station to a node. In this paper, we consider the H-max-hop routing strategy, in which, if the destination is located within H (H ≥ 1) hops from the source node, data packets are transmitted in ad hoc communications. Otherwise, data packets are forwarded in infrastructure communications. The base station then relays the packets through the wired network. After the packets arrive at the base station that is closest to the destination node, the base station then forwards the packets to the destination node (i.e., the downlink infrastructure communication). It is obvious that when there is an uplink communication, there is always a downlink communication. We then divide the total bandwidth of W bits/sec into three parts: (1) WA for ad hoc communications, (2) WI,U for uplink infrastructure communications and (3) WI,D for downlink infrastructure communications. Since WI,U is equal to WI,D , it is obvious that W = WA + WI,U + WI,D = WA + 2WI,U . To simplify our analysis, we use WI to denote either WI,U or WI,D . TABLE I T HE MAIN RESULTS = = = = = Conditions Per-node Throughput λ Connectivity Θ( HCWA ) log n A Condition = = = = Interference = Θ( Condition Fig. 2. Interface-bottleneck All possible sub-cases considered Condition Corresponding to the partition of the bandwidth, we also split the C channels into two disjoint groups CA and CI , in which CA channels are dedicated for ad hoc communications and CI channels are dedicated for infrastructure communications. Thus, C = CA + CI . Besides, each base station is mounted with m NICs, which serve for both the uplink traffic and the downlink traffic. It is obvious that the number of NICs serving for the uplink traffic is equal to the number of NICs serving for the downlink traffic. So, m must be an even number. To the best of our knowledge, we are the first to propose such an MC-IS network, which has not been studied in the literature. A. Contributions and Main Results The primary research contributions of our paper are summarized as follows. (1) We formally identify an MC-IS network that characterizes the features of multi-channel wireless networks with infrastructure support. The capacity and the average delay of an MC-IS network have not been studied before. (2) We derive both the upper bounds and the constructive lower bounds of the capacity of an MC-IS network contributed by ad hoc communications. Importantly, the orders of the lower bounds are the same as the orders of the upper bounds, meaning that the upper bounds are tight. We also derive the delay of an MC-IS network contributed by ad hoc communications. (3) We found that the capacity of an MC-IS network contributed by ad hoc communications is mainly limited by four requirements - connectivity requirement, interference requirement, destination-bottleneck requirement and interface-bottleneck requirement. Regarding (1), we identify the characteristics of an MCIS network and describe the network topology, the network communications and the routing strategy in Section II, which also presents the models and assumptions used in this paper. We then derive the upper bounds on the network capacity contributed by ad hoc communications in Section III and the constructive lower bounds in Section IV, both of which bring us to (2) above. With regard to (3), we summarize our main results in Table I, which are stated in Theorem 1 and Theorem 2. Specifically, WA 1 1 ) 2 H log 2 n CA Condition Destination-bottleneck Delay D Θ(H) 1 Θ( n 2 log log(H 2 log n)WA ) 1 CA H log 2 n·log(H 2 log n) n · Θ(H 2 log n WA CA ) we found that the capacity of an MC-IS network is mainly limited by four requirements: (i) Connectivity requirement the need to ensure that the network is connected so that each source node can successfully communicate with its destination node; (ii) Interference requirement - two receivers simultaneously receiving packets from two different transmitters must be separated with a minimum distance to avoid the interference; (iii) Destination-bottleneck requirement - the maximum amount of data that can be simultaneously received by a destination node; (iv) Interface-bottleneck requirement - the maximum amount of data that an interface can simultaneously transmit or receive. We also found that each of the four requirements dominates the other three requirements in terms of the throughput of the network under different conditions on CA and H. Specifically, CA can be partitioned into 3 cases: (1) the case when CA = O(F1 ), (2) the case when CA = Ω(F1 ) and CA = O(F2 ), and (3) the case when CA = Ω(F2 ), where 2 log n) 2 F1 = log n and F2 = n( logloglog(H(H 2 log n) ) . Under each of the above cases, H can be partitioned into two sub-cases. Under the first case, H is partitioned into 2 sub-cases, namely Sub-case 1 (when H = o(G1 )) and Sub2 1 case 2 (when H = Ω(G1 )), where G1 = n 3 / log 3 n. Under the second case, H is partitioned into 2 sub-cases, namely Sub-case 3 (when H = o(G2 )) and Sub-case 4 (when H = 1 1 1 Ω(G2 )), where G2 = n 3 CA6 / log 2 n. Under the third case, H is partitioned into 2 sub-cases, namely Sub-case 5 (when H = o(G3 )) and Sub-case 6 (when H = Ω(G3 )), where G3 = 1 1 n 2 / log 2 n. Fig. 2 shows all possible sub-cases. Specifically, each requirement dominates the other at least one sub-case under different conditions as follows. • • • • Connectivity Condition: corresponding to Sub-case 2 in which Connectivity requirement dominates. Interference Condition: corresponding to Sub-case 4 in which Interference requirement dominates. Destination-bottleneck Condition: corresponding to Subcase 6 in which Destination-bottleneck requirement dominates. Interface-bottleneck Condition: corresponding to Subcase 1, Sub-case 3, or Sub-case 5, in which Interfacebottleneck requirement dominates. II. F ORMULATION AND M ODELS A. Interference model We consider the interference model [1]–[6]. When node X1 transmits to node X2 over a particular channel, the transmission is successfully completed by node X2 if no node within the transmission range of X2 transmits over the same channel. Therefore, for any other node X3 simultaneously transmitting over the same channel, and any guard zone ∆ > 0, the following condition holds. dist(X3 , X2 ) ≥ (1 + ∆)dist(X1 , X2 ) where dist(X1 , X2 ) denotes the distance between two nodes X1 and X2 . The interference model applies for both ad hoc communications and infrastructure communications. Since ad hoc communications and infrastructure communications are separated by different channels (i.e., CA and CI ), the interference only occurs either between two ad hoc communications or between two infrastructure communications. B. Definitions of Throughput Capacity and Delay Definition 1: Feasible per-node throughput. For an MC-IS network, a throughput of λ (in bits/sec) is feasible if by ad hoc communications or infrastructure communications, there exists a spatial and temporal scheme, within which each node can send or receive λ bits/sec on average. Definition 2: Per-node throughput capacity with the throughput of λ is of order Θ(g(n)) bits/sec if there are deterministic constants h > 0 and h′ < +∞ such that limn→∞ limn→∞ inf P (λ = hg(n) is feasible) = 1 and P (λ = h′ g(n) is feasible) < 1 Besides, we use T and TA to denote the feasible aggregate throughput and the feasible aggregate throughput contributed by ad hoc communications, respectively. The delay of a packet D is defined as the time that it takes for the packet to reach its destination after it leaves the source [7]. Averaging the delay of all the packets transmitted in the whole network, we obtain the average delay of a network. III. U PPER B OUNDS ON N ETWORK C APACITY C ONTRIBUTED BY A D H OC C OMMUNICATIONS We first derive the upper bounds on the per-node throughput capacity under Connectivity Condition. Proposition 1: When Connectivity requirement dominates, the per-node throughput capacity contributed by ad hoc comA ). munications is λa = O( H 3 CnW 2 A log n Proof. We first calculate the expectation of the number of hops under the H-max-hop routing scheme, which is denoted by h h = E(h) = 1 · P (h = 1) + 2 · P (h = 2) + . . . +H · P (h = H) 3πr2 (n) πr2 (n) +2· + ... = 1· 2 2 πH r (n) πH 2 r2 (n) (H 2 − (H − 1)2 )πr2 (n) +H · πH 2 r2 (n) 4H 3 + 3H 2 − H = (1) 6H 2 where P (h = i) (i = 1, 2, . . . , H) is the probability that a packet traverses h = i hops. From Eq. (1), we have h ∼ H. We then calculate the probability that a node uses the ad hoc mode to transmit, denoted by P (AH), which is the probability that the destination node is located within H hops away from the source node. Thus, we have P (AH) = πH 2 r2 (n) (2) Since each source generates λa bits per second and there are totally n sources, the total number of bits per second served by the whole network on a particular channel is required to be at A least n·P (AH)·h·λa , which is bounded by Nmax · W CA , where Nmax is the maximum number of simultaneous transmissions on any particular channel, which is upper bounded by Nmax ≤ k1 ∆2 (r(n))2 (k1 > 0 is a constant, independent of n) [1]. Then, A we have n · P (AH) · h · λa ≤ Nmax · W CA . Combining the above results yields: λa ≤ k1 2 ∆ r2 (n) · WA 3 nπH r2 (n)CA ≤ k2 WA 3 nH r2 (n)CA where k2 is a constant. Besides, to guarantee that the network is p connected with high probability (w.h.p.)1, we require r(n) > log n/πn [1]. 3 nWA Thus, we have λa ≤ H 3klog 2 nC , where k3 is a constant. A We then derive the upper bounds on the per-node throughput capacity under Interference Condition. Proposition 2: When Interference requirement dominates, the per-node throughput capacity contributed by ad hoc communications is λa = O( 1 nWA 3 ). CA2 H 3 log 2 n Proof. When Interference Condition is satisfied, the per-node throughput is limited by the interference requirement [1]. Thus, we can use the theorem derived under arbitrary networks [1]. Similarly, we assume that all nodes are synchronized. Let the average distance between a source and a destination be l, which is roughly bounded by h · r(n). In the network with n nodes and under the H-max-hop routing scheme, there are at most n · P (AH), where P (AH) is the probability that a node transmits in ad hoc mode and can be calculated by Eq. (2). Within any time period, we consider a bit b, 1 ≤ b ≤ λnP (AH) We assume that bit b traverses h(b) 1 We say that an event e happens with a high probability if P (e) → 1 when n → ∞. hops on the path from the source to the destination, where the h-th hop traverses a distance of r(b, h). It is obvious that the distance traversed by a bit from the source to the destination is no less than the length of the line jointing the source and the destination. Thus, after summarizing the traversing distance of all bits, we have nλa P (AH) h(b) λa · nl · P (AH) ≤ X X b=1 r(b, h) h=1 Let Th be the total number of hops traversed by all bits in a Pnλ P (AH) second and we have Th = b=1a h(b). Since each node A has one interface which can transmit at most W CA , the total number of bits that can be transmitted by all nodes over all An interfaces are at most W 2CA , i.e., Th ≤ WA n 2CA (3) On the other hand, under the interference model, we have the following in-equation from [1] ∆ (dist(X3 − X4 ) + dist(X1 − X2 )) 2 where X1 and X3 denote the transmitters and X2 and X4 denote the receivers. This in-equation implies that each hop consumes a disk of radiums ∆ 2 times the length of the hop. Therefore, we have dist(X1 − X2 ) ≥ nλa P (AH) h(b) X b=1 X π∆2 (r(b, h))2 ≤ WA 4 nλa P (AH) h(b) b=1 X 1 4WA (r(b, h))2 ≤ Th π∆2 Th (4) h=1 Since the left hand side of this in-equation is convex, we have nλa P (AH) h(b) X X 1 X 1 2 r(b, h)) ≤ (r(b, h))2 (5) Th Th nλa P (AH) h(b) ( X b=1 b=1 h=1 h=1 Joining (4)(5), we have nλa P (AH) h(b) X b=1 X r(b, h) ≤ h=1 r 4WA Th π∆2 From (3), we have nλa P (AH) h(b) X b=1 X h=1 r(b, h) ≤ WA r λa ≤ q log n πn , we have k4 nWA 1 2 3 CA H 3 log 2 n Before deriving the upper bounds on the throughput capacity under the destination-bottleneck condition, we need to bound the number of flows towards a node under the H-max-hop routing scheme. Specifically, we have the following result. Lemma 1: The maximum number of flows towards a node under the H-max-hop routing scheme is DH (n) = 2 log n) Θ( loglog(H log(H 2 log n) ) with high probability (w.h.p.). Proof. As shown in [6], the total number of source nodes transmitting in ad hoc mode under the H-max-hop routing scheme is Θ(H 2 log n) w.h.p.. Besides, it is proved in [8] that the maximum number of flows towards any given node in a random ad hoc network with n nodes is upper bounded by Θ( logloglogn n ) w.h.p.. Combining the two results leads to the above result. We then obtain the upper bounds on the per-node throughput capacity under Destination-bottleneck Condition. Proposition 3: When Destination-bottleneck requirement dominates, the per-node throughput capacity contributed by ad hoc communications is λa = 3 3 O(n 2 log log(H 2 log n)WA /(CA H 3 log 2 n · log(H 2 log n))). Proof. Each node has one interface that can support at most WA CA . Since each node has at most DH (n) flows under the H-max-hop routing scheme, the data rate of the minimum A rate flow is at most CAW DH (n) , where DH (n) is bounded 2 h=1 This in-equation can be rewritten as X Since r(n) > 2n π∆2 CA Pnλ P (AH) Ph(b) Besides, since λa ·nl·P (AH) ≤ b=1a h=1 r(b, h), we have q q q 2n 2 WA π∆2n W W A A 2C π∆2 CA π∆2 nCA A λa ≤ = ≤ πH 3 (r(n))3 nl · P (AH) nhr(n)πH 2 (r(n))2 log n) by Θ( loglog(H log(H 2 log n) ) by Lemma 1. After calculating all the data rates at each node times with the traversing distance, we have n · P (AH) · λA · h · r(n) ≤ CAWDAHn(n) · 1, where P (AH) and h are defined in the proof of PropoWA sition 1. We then have λA ≤ C D (n)P ≤ (AH)hr(n) A H WA CA πH 3 r 3 (n)·log(H 2 log n)/ log log(H 2 log n) 2 2 since h ∼ H and P (AH) = πH r (n) as shown in the proof of Proposition p 3 1. Since r(n) = Θ( log n/n) [1], we have λA ≤ WA n 2 · 3 log log(H 2 log n)/(CA H 3 log 2 n · log(H 2 log n)) Finally, we prove the upper bounds on the per-node throughput capacity under Interface-bottleneck Condition. Proposition 4: When Interface-bottleneck requirement dominates, the per-node throughput capacity contributed by A ad hoc communications is λa = O( W CA ). Proof. In an MC-IS network, each node is equipped with only A one NIC supporting at most W CA data rate. Thus, λa is also WA upper bounded by CA for any network settings. IV. C ONSTRUCTIVE L OWER B OUNDS ON N ETWORK C APACITY C ONTRIBUTED BY A D H OC C OMMUNICATIONS We first divide the plane into a number of equal-sized cells. The size of each cell is properly chosen so that each cell has Θ(na(n)) nodes, where a(n) is the area of a cell. We then design a routing scheme to assign the number of flows at each node evenly. Finally, we design a Time Division Multiple Access (TDMA) scheme to schedule the traffic at each node. − Fig. 4. Fig. 3. Plane divided into a number of cells and each with area a(n). A. Cell Construction We divide the plane into 1/a(n) equal-sized cells and each cell is a square with area of a(n), as shown in Fig. 3. The cell size of a(n) must be carefully chosen to fulfill the three requirements, i.e., Connectivity requirement, Interference requirement and Destination-bottleneck requirement. We set a(n) = √ min(max( 100 3 CA log n , log 2 n 1 n/CA2 n), log3 3 2 n·log(H 2 log n) n 2 ·log log(H 2 log n) ) similar to [4]. Note that Interface-bottleneck requirement is independent of the size of a cell. The maximum number of nodes in a cell is bounded by the following lemma. n , then each cell has Lemma 2: [4] If a(n) > 50 log n Θ(n(a(n)) nodes w.h.p.. We next check whether all the above values of a(n) are properly chosen such that each cell has Θ(n(a(n)) nodes w.h.p. when n is large enough (i.e., Lemma 2 is satisfied). It √ 1 3 n is obvious that 100 CnA log n > 50 log and log 2 n/(CA2 n) > n 50 log n (as we only consider CA in Connectivity Condition and n 3 Interference Condition). Besides, n greater than 50 log with n 3 3 n and log 2 n/n 2 > 50 log n large n log 2 n·log(H 2 log n) 3 n2 is also ·log log(H 2 log n) 2 log n) since loglog(H log(H 2 log n) > 1 when n is large enough. It is also proved in [7], [9] that the number of interfering cells around a cell is bounded by a constant k5 , which is independent of n. B. Routing Scheme To assign the flows at each node evenly, we design a routing scheme consists of two steps: (1) Assigning sources and destinations and (2) Assigning the remaining flows in a balanced way. In Step (1), each node is the originator of a flow and each node is the destination of at most DH (n) flows, where DH (n) is defined in Lemma 1. Thus, after Step (1), there are at most 1 + DH (n) flows. We denote the straight line connecting a source S to its destination D as an S-D lines. In Step (2), we need to calculate the number of S-D lines (flows) passing through a cell so that we can assign them to each node evenly. Specifically, we have the following result. Lemma 3: The number of S-D lines passing through a cell is bounded by O(nH 3 (a(n))2 ). Proof. We present a proof of the bound in [9]. TDMA transmission schedule As shown in Lemma 2, there are Θ(n · a(n)) nodes in each cell. Therefore, Step (2) will assign to any node at most 3 (a(n))2 ) = O(H 3 a(n)) flows. Summarizing Step (1) O( nHn·a(n) and Step (2), there are at most f (n) = O(1 + H 3 a(n) + DH (n)) flows at each node. On the other hand, H 3 a(n) dominates f (n) since H > 1 and a(n) is asymptotically larger than DH (n) when n is large enough. Thus, we have f (n) = O(H 3 a(n)). C. Scheduling Transmissions We next design a scheduling scheme to transmit the traffic flows assigned in a routing scheme. Any transmissions in this network must satisfy the two additional constraints simultaneously: 1) each interface only allows one transmission/reception at the same time, and 2) any two transmissions on any channel should not interfere with each other. We propose a TDMA scheme to schedule transmissions that satisfy the above two constraints. Fig. 4 depicts a schedule of transmissions on the network. In this scheme, one second is divided into a number of edge-color slots and at most one transmission/reception is scheduled at every node during each edge-color slot. So, the first constraint is satisfied. Each edgecolor slot can be further split into smaller mini-slots. In each mini-slot, each transmission satisfies the above two constraints. Then, we describe the two time slots as follows. (i) Edge-color slot: First, we construct a routing graph in which vertices are the nodes in the network and an edge denotes transmission/reception of a node. In this construction, one hop along a flow is associated with one edge in the routing graph. In the routing graph, each vertex is assigned with f (n) = O(H 3 a(n)) edges, which can be edge-colored with at most O(H 3 a(n)) colors [4], [10]. We then divide one second into O(H 3 a(n)) edge-color slots, each of which has a length 1 of Ω( H 3 a(n) ) seconds and is stained with a unique edge-color. Since all edges connecting to a vertex use different colors, each node has at most one transmission/reception scheduled in any edge-color time slot. (ii) Mini-slot: We further divide each edge-color slot into mini-slots. Then, we build a schedule that assigns a transmission to a node in a mini-slot within an edge-color slot over a channel. We construct an interference graph in which each vertex is a node in the network and each edge denotes the interference between two nodes. We then show as follows that the interference graph can be vertex-colored with k7 (na(n)) colors, where k7 is a constant defined in [4]. Lemma 4: The interference graph can be vertex-colored with at most O(na(n)) colors. Proof. We present the detailed proof in [9]. We need to schedule the interfering nodes either on different channels, or at different mini-slots on the same channel since two nodes assigned the same vertex-color do not interfere with each other, while two nodes stained with different colors may interfere with each other. We divide each edge-color slot into ⌈k7 na(n)/CA ⌉ mini-slots on every channel, and assign the mini-slots on each channel from 1 to ⌈k7 na(n)/CA ⌉. A node assigned with a color s, 1 ≤ s ≤ k7 na(n), is allowed to transmit in mini-slot ⌈s/CA ⌉ on channel (s mod CA ) + 1. We next have the constructive lower bounds of the capacity. Proposition 5: The achievable per-node throughput capacity λa contributed by ad hoc communications is as follows. 1) When Connectivity requirement dominates, λa is A Ω( H 3 CnW ) bits/sec; 2 A log n 2) When Interference requirement dominates, λa is nWA Ω( ) bits/sec; 1 3 H 3 CA2 log 2 n 3) When Destination-bottleneck requirement dominates, λa 3 is Ω( n 2 log log(H 2 log n)WA 3 CA H 3 log 2 n·log(H 2 log n) ) bits/sec; 4) When Interface-bottleneck requirement dominates, λa is A Ω( W CA ). 1 Proof. Since each edge-color slot with a length of Ω( H 3 a(n) ) m l k7 na(n) mini-slots over every chanseconds is divided into CA m l 1 nel, each mini-slot has a length of Ω(( H 3 a(n) ) )/ k7 na(n) CA A seconds. Since each channel can transmit at the rate of W CA WAl m bits/sec, in each mini-slot, λa = Ω( k na(n) ) (CA H 3 a(n)· 7 C A l m k7 na(n) bits can be transported. Since k7 na(n) ≤ + 1, CA CA WA we have, λa = Ω( k7 H 3 a2 (n)n+H 3 a(n)C ) bits/sec. Thus, A WA WA λa = Ω(M INO ( H 3 a2 (n)n , H 3 a(n)C )) bits/sec (where A M INO (f (n), g(n)) is equal to f (n) if f (n) = O(g(n)); otherwise it is equal to g(n)). Recall that a(n) is set to 1 min(max( 3 3 2 2 100CA2 log n log 2 n , 1 ), log3 n·log(H 2 log n) ). n n 2 ·log log(H log n) CA2 n Substituting the three values to λa , we have the results 1), 2) and 3). Besides, each interface can transmit or receive WA A at the rate of W CA bits/sec. Thus, λa = Ω( CA ), which is the result 4). D. Summary It is shown in [6] that the total traffic of ad hoc communications is nπH 2 r2 (n)λA . Combining Propositions 1, 2, 3, 4 and 5 leads to the following theorem. Theorem 1: The aggregate throughput capacity of the network contributed by ad hoc communications is 1) When Connectivity requirement dominates, TA is A ) bits/sec. Θ( HCnW A log n 2) When Interference requirement dominates, TA is Θ( 1 nWA 1 ) bits/sec. 2 H log 2 n CA 3) When Destination-bottleneck requirement dominates, 3 TA is Θ( n 2 log log(H 2 log n)WA 1 CA H log 2 n·log(H 2 log n) ) bits/sec. 4) When Interface-bottleneck requirement dominates, TA is A Θ(H 2 log n · W CA ) bits/sec. We then derive the average delay of an MC-IS network and have the following result. Theorem 2: Under the H-max-hop ad hoc routing strategy, if the packets are transmitted in the ad hoc mode and along a route which approximates the straight line connecting the source and the destination, the average delay is Θ(H). Proof. The average delay of the packets transmitted in the ad hoc mode under the H-max-hop routing strategy in an SC-IS network is bounded by Θ(H) [6], which also holds for an MC-IS network since both an SC-IS network and an MC-IS network have the same routing strategy. V. C ONCLUSION In this paper, we propose a novel multi-channel wireless network with infrastructure (named an MC-IS network), which consists of common nodes, each with a single interface, and infrastructure nodes, each with multiple interfaces. We derive the upper bounds and lower bounds on the capacity of an MC-IS network contributed by ad hoc communications, where the upper bounds are proved to be tight. We also prove that the average delay contributed by ad hoc communications is bounded by H, which is the maximum number of hops in H-max routing scheme. There are some interesting questions in this new type of networks: (1) what are the upper bounds on the capacity of an MC-IS network contributed by infrastructure communications? (2) are the upper bounds also tight? (3) what is the average delay of an MC-IS network with considering both ad hoc communications and infrastructure communications? To solve the above questions would be one of our future works. VI. ACKNOWLEDGMENT The work described in this paper was supported by Macao Science and Technology Development Fund under Grant No. 036/2011/A and Grant No. 081/2012/A3. The authors would like to thank Gordon G.-D. Han for his constructive comments. R EFERENCES [1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 388 – 404, 2000. [2] B. Liu, Z. Liu, and D. Towsley, “On the Capacity of Hybrid Wireless Networks,” in Proc. of IEEE INFOCOM, 2003. [3] U. C. Kozat and L. 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