8 such a feasible retrieval pair is thus at least.It follows average placement delay.Therefore,it is not clear how the that the expected time of reaching a client of content ik is cache size affects the average weighted sum delay,and we are upper bounded by which are the same for all i,after thus motivated to investigate how the network performance plugging in the values of Mi in (21)and Ri in(24).Then,it scales with the cache size no. is easy to show that,each content ik can reach all its clients We first study the average weighted sum delay cost C under 9(②∑)reg四 the fast mobility model.The average weighted sum delay cost within timeslots. Vn C in the following theorem is an extension of the average Combining all the analysis above,we can conclude this delay cost W in Theorem 1. theorem. ◆ Theorem 5:Under the fast mobility model and Assumption 1,for any scheduling scheme with per-node cache size no.the IV.PLACEMENT COST AND CACHE CONSTRAINTS average weighted sum delay cost of serving all nK requests satisfies In this section,we further study the asymptotic delay performance under additional constraints on two important VpK2A2 logn %=P(4)手)，P≠0 factors:cache size and placement cost.The new results are not KA otherwise only more general than the previous results,but also provide √nb log n new insights into the impacts of different network parameters (27) on the network performance. where A=∑1V. From the previous analysis,we can see that the placement Proof:Please refer to Appendix C. ■ cost is captured in the average delay cost.However,one Next,we study the average weighted sum delay cost C concern is that the unit of the placement cost is equal to under the slow mobility model.The average weighted sum that of the retrieval cost,which may not be the case in delay cost C in the following theorem is an extension of the reality.For example,there are peak-traffic time and off-peak average delay cost W in Theorem 3. time.Content placement can take place at off-peak time to Theorem 6:Under the slow mobility model and Assump- reduce the data traffic at peak time.Therefore,the units of tion 1,for any scheduling scheme with per-node cache size the delay costs in the placement and retrieval phases could be nb,the average weighted sum delay cost of serving all nk different in general,and the unit cost for the time consumed requests satisfies in the placement phase may be lower than that in the retrieval phase.These practical concerns motivate us to incorporate the （p() n6=2( KDogn 0 ,p≠0 following assumption in studying content-centric MANETs. KD Assumption 1:The delay cost of a placement timeslot is otherwise lower than that of a retrieval timeslot. (28) We normalize the delay cost of a retrieval timeslot to 1,and where D= denote the delay cost of a placement timeslot as p.Here,p =1 can be treated as the delay cost weight.By Assumption 1,we Proof:Please refer to Appendix D. ■ know that 0<p<1.For a given request state Q,denote the The results in Theorem 5 and Theorem 6 are general and time length of the placement phase and the retrieval phase as hold for any n and p.In particular,the result in Theorem 5 WP.and WR.Q,respectively.Then,for a given request state can cover two special cases under fast mobility,i.e.,content Q,the average weighted sum delay cost of serving all nK placement is free (p=0)[17]and the unit of the placement requests can be expressed as cost is the same as that of the retrieval cost (p 1)in Theorem 1.The result in Theorem 6 can cover the one in Co pWP.Q+WR.Q. (25) Theorem 3(p 1)under slow mobility.Lower bounds on The average weighted sum delay cost C of serving all nK the average weighted sum delay cost C for different cases requests over all possible request states is under fast and slow mobility are presented in Table I and Table II,respectively.The insights into these cases are provided as C≌ECal=CoP(Q): (26) follows. QEX Case 1:When the cost of content placement is not con- where P(O)is the probability of being state O. sidered,i.e.,p =0,the average weighted sum delay cost Moreover,in reality,caches always have limited sizes, solely comes from content retrieval.The total (retrieval)cost which is a main constraint for content placement.Therefore, decreases with the cache size n. we consider that each node is equipped with a cache of the Case 2:When p0 and the cache size is smaller than a same size.Let nb be the cache size of each node,measured threshold,as n increases,the placement cost increases and the in the number of contents it can store.Denote np as the sum retrieval cost decreases.Moreover,the larger the cache size, of the occupied cache memory over all nodes during content the smaller the average weighted sum delay cost. placement.Then,we have np nnb,which sets a constraint Case 3:When p0 and the cache size is larger than on the sum of replicas in the placement phase.Intuitively, a threshold,the placement cost is on the same order of the when the cache size is larger,more replicas can be cached. retrieval cost.In this case,the minimum average weighted sum which results in a smaller average retrieval delay and a larger delay cost)()is achieved under8 such a feasible retrieval pair is thus at least MiR 2 i 9 . It follows that the expected time of reaching a client of content ik is upper bounded by 9 MiR2 i , which are the same for all i, after plugging in the values of Mi in (21) and Ri in (24). Then, it is easy to show that, each content ik can reach all its clients within 9 €PI i=1 PK k=1 n 2 3 i Š3/4 (log n) 5 4 √ n timeslots. Combining all the analysis above, we can conclude this theorem. IV. PLACEMENT COST AND CACHE CONSTRAINTS In this section, we further study the asymptotic delay performance under additional constraints on two important factors: cache size and placement cost. The new results are not only more general than the previous results, but also provide new insights into the impacts of different network parameters on the network performance. From the previous analysis, we can see that the placement cost is captured in the average delay cost. However, one concern is that the unit of the placement cost is equal to that of the retrieval cost, which may not be the case in reality. For example, there are peak-traffic time and off-peak time. Content placement can take place at off-peak time to reduce the data traffic at peak time. Therefore, the units of the delay costs in the placement and retrieval phases could be different in general, and the unit cost for the time consumed in the placement phase may be lower than that in the retrieval phase. These practical concerns motivate us to incorporate the following assumption in studying content-centric MANETs. Assumption 1: The delay cost of a placement timeslot is lower than that of a retrieval timeslot. We normalize the delay cost of a retrieval timeslot to 1, and denote the delay cost of a placement timeslot as ρ. Here, ρ can be treated as the delay cost weight. By Assumption 1, we know that 0 ≤ ρ ≤ 1. For a given request state Q, denote the time length of the placement phase and the retrieval phase as W¯ P,Q and W¯ R,Q, respectively. Then, for a given request state Q, the average weighted sum delay cost of serving all nK requests can be expressed as C¯Q = ρW¯ P,Q + W¯ R,Q. (25) The average weighted sum delay cost C¯ of serving all nK requests over all possible request states is C¯ , E[C¯Q] = X Q∈A C¯QP(Q), (26) where P(Q) is the probability of being state Q. Moreover, in reality, caches always have limited sizes, which is a main constraint for content placement. Therefore, we consider that each node is equipped with a cache of the same size. Let nb be the cache size of each node, measured in the number of contents it can store. Denote np as the sum of the occupied cache memory over all nodes during content placement. Then, we have np ≤ nnb, which sets a constraint on the sum of replicas in the placement phase. Intuitively, when the cache size is larger, more replicas can be cached, which results in a smaller average retrieval delay and a larger average placement delay. Therefore, it is not clear how the cache size affects the average weighted sum delay, and we are thus motivated to investigate how the network performance scales with the cache size nb. We first study the average weighted sum delay cost C¯ under the fast mobility model. The average weighted sum delay cost C¯ in the following theorem is an extension of the average delay cost W¯ in Theorem 1. Theorem 5: Under the fast mobility model and Assumption 1, for any scheduling scheme with per-node cache size nb, the average weighted sum delay cost of serving all nK requests satisfies C¯ ≥ 8 < : Θ √3 ρK2A2 log n ‹ , nb = Ω€€ KA ρ Š 2 3 Š , ρ ̸= 0 Θ € √ KA nb log n Š , otherwise (27) where A = PI i=1 √pi . Proof: Please refer to Appendix C. Next, we study the average weighted sum delay cost C¯ under the slow mobility model. The average weighted sum delay cost C¯ in the following theorem is an extension of the average delay cost W¯ in Theorem 3. Theorem 6: Under the slow mobility model and Assump￾tion 1, for any scheduling scheme with per-node cache size nb, the average weighted sum delay cost of serving all nK requests satisfies C¯ ≥ 8 >< >: Θ  ρ 1 4 € KD log n Š 3 4 ‹ , nb = Ω€€ KD√3 log n ρ Š 3 4 Š , ρ ̸= 0 Θ  KD √3 nblog2n ‹ , otherwise (28) where D = P I i=1 p 2 3 i . Proof: Please refer to Appendix D. The results in Theorem 5 and Theorem 6 are general and hold for any nb and ρ. In particular, the result in Theorem 5 can cover two special cases under fast mobility, i.e., content placement is free (ρ = 0) [17] and the unit of the placement cost is the same as that of the retrieval cost (ρ = 1) in Theorem 1. The result in Theorem 6 can cover the one in Theorem 3 (ρ = 1) under slow mobility. Lower bounds on the average weighted sum delay cost C¯ for different cases under fast and slow mobility are presented in Table I and Table II, respectively. The insights into these cases are provided as follows. Case 1: When the cost of content placement is not con￾sidered, i.e., ρ = 0, the average weighted sum delay cost solely comes from content retrieval. The total (retrieval) cost decreases with the cache size nb. Case 2: When ρ ̸= 0 and the cache size is smaller than a threshold, as nb increases, the placement cost increases and the retrieval cost decreases. Moreover, the larger the cache size, the smaller the average weighted sum delay cost. Case 3: When ρ ̸= 0 and the cache size is larger than a threshold, the placement cost is on the same order of the retrieval cost. In this case, the minimum average weighted sum delay cost Θ €√3 ρK2A2 log n Š (Θ € ρ 1 4 € KD log n Š 3 4 Š ) is achieved under