Controlled mobility in stochastic and dynamic wireless networks

Abstract

We consider the use of controlled mobility in wireless networks where messages arriving randomly in time and space are collected by mobile receivers (collectors). The collectors are responsible for receiving these messages via wireless transmission by dynamically adjusting their position in the network. Our goal is to utilize a combination of wireless transmission and controlled mobility to improve the throughput and delay performance in such networks. First, we consider a system with a single collector. We show that the necessary and sufficient stability condition for such a system is given by ρ<1 where ρ is the expected system load. We derive lower bounds for the expected message waiting time in the system and develop policies that are stable for all loads ρ<1 and have asymptotically optimal delay scaling. We show that the combination of mobility and wireless transmission results in a delay scaling of $\varTheta(\frac{1}{1-\rho})$ with the system load ρ, in contrast to the $\varTheta(\frac{1}{(1-\rho)^{2}})$ delay scaling in the corresponding system without wireless transmission, where the collector visits each message location. Next, we consider the system with multiple collectors. In the case where simultaneous transmissions to different collectors do not interfere with each other, we show that both the stability condition and the delay scaling extend from the single collector case. In the case where simultaneous transmissions to different collectors interfere with each other, we characterize the stability region of the system and show that a frame-based version of the well-known Max-Weight policy stabilizes the system asymptotically in the frame length.

Appendices

Appendix A: Proof of Theorem 1

Let N(t) denote the number of messages in the system at time t, and let W _j denote the delay experienced by the jth message. Recall the definition of time $t_{k},\;k\ge1$, the time at which the collector returns to the center of the network region for the kth time, where t ₀≐0. Let N(t _k ) denote the total number of messages waiting for service at time t _k . We will denote N(t _k ) by N _k for notational simplicity. The duration of time between t _k−1 and t _k is called the kth cycle, and is denoted by C _k , k∈ℤ₊. Note that $\{N_{k}:\; k \in \mathbb{N}\}$ is an irreducible Markov chain on countable state space ℕ, termed the cycle Markov chain. Given the system state N _k at time t _k , we find the TSPN tour of length L _k through the N _k neighborhoods.⁵ We prove Theorem 1 by establishing the following properties:

• 1. We first prove that the discrete-time Markov chain {N _k } is positive recurrent and has a steady state distribution with a finite first moment.

• 2. Using this steady state distribution, we derive bounds on the first and second moments of the cycle duration, as well as the residual and past cycle durations under the TSPN policy.

• 3. Next, we show that the message delays {W _j :j∈ℤ₊} and the queue length process {N(t):t≥0} form positive recurrent regenerative processes and, therefore, converge in distribution to stationary processes.

• 4. Using the bounds on the residual and the past cycle times, we show that the stationary process of message delays has a finite expectation.

• 5. Finally, we utilize the stationary version of Little’s law to show that the stationary process of number of messages in the system has finite expectation.

Cycle Markov chain {N _k }

First, we will use the Foster–Lyapunov criterion to show that the Markov chain described by the states N _k is positive recurrent. We use the linear Lyapunov function V(N _k )=sN _k , the total load served during the kth cycle. Note that V(0)=0, S _k ={x:V(x)≤B} is a bounded set for all finite B and V(.) is a nondecreasing function. Since the arrival process is Poisson, the expected number of arrivals during a cycle can be upper bounded as follows:

$$ \mathbb{E} [N_{k+1}| N_k] \le \lambda (L/v + sN_k ). $$

(18)

Hence, we obtain the following drift expression for the load during a cycle:

$$ \mathbb{E} [sN_{k+1}- sN_k| N_k] \le \rho L/v - (1-\rho) sN_k. $$

(19)

Since ρ<1, there exist a δ>0 such that ρ+δ<1:

$$ \mathbb{E} [s N_{k+1}- s N_k| N_k] \le \rho L/v - \delta sN_k. $$

Fix ϵ∈(0,δ). A simple derivation shows that when N _k is outside the finite and bounded set $S = \{N \in \mathbb{N}: N \le \frac{\rho L/v+\epsilon}{s(\delta-\epsilon)} \}$ the drift expression is given by

$$ \mathbb{E} [s N_{k+1}- s N_k| N_k] \le - \epsilon(1+ sN_k). $$

For N _k ∈S, using ρ<1−δ and the definition of the set S, we have from (18),

$$ \mathbb{E} [s N_{k+1}| N_k] \le \rho L/v + (1- \delta) sN_k <\rho L/v + \frac{(1-\delta)(\rho L/v+\epsilon)}{(\delta-\epsilon)} <\infty. $$

Moreover, since the state space is countable, the set S is finite, and since the states in the Markov chain {N _k } have nonzero probability of self transition, the Markov chain is strongly aperiodic. Therefore, all the conditions of Lemma 4.2 in [2] are satisfied (by the choice of the function g(N _k )=1+sN _k ), and we have that the Markov chain {N _k } is positive (Harris) recurrent, N _k has a steady state distribution, where we let the random variable N ^c denote this steady state distribution. Moreover, $\mathbb{E}[N_{k}]$ converges to $\mathbb{E}[N^{c}]$, and the expected number of messages in steady state, $\mathbb{E}[N^{c}]$, is finite [2].

Moments of cycle duration

Next, we derive bounds on the first and second moments of the cycle duration, and the expected residual and past cycle durations. These bounds will be necessary in order to obtain an upper bound for the expected message delay and the number of messages in the system. We will prove the finiteness of the expected number of messages in the system by first establishing that the expected message delay in the system is finite, and then utilizing the stationary version of Little’s law [55]. The analysis in this section is similar to that in [2]. Let C _k denote the duration of the kth cycle, C _k =sN _k +L _k /v. The location distributions of messages in different cycles are independent and uniformly distributed and the TSPN policy obtains the travel paths, L _k , using a stationary algorithm [41]. Therefore, C _k is a function of N _k and the location distribution of these N _k messages. Note that the lengths of the travel paths L _k are uniformly bounded from above by L for all k∈ℤ₊. Let $\mathbb{E}^{0}$ denote expectation at the time corresponding to the beginning of a cycle, in steady state. We let N ^c and C denote the steady state versions of N _k and C _k . Taking the expectation of (18) with respect to the steady state distribution at the beginning of the cycles we have

$$ \mathbb{E}^0 \bigl[N^c\bigr] \le \frac{\lambda L}{v} + \rho \mathbb{E}^0\bigl[N^c\bigr], $$

which implies that

$$ \mathbb{E}^0 \bigl[N^c\bigr] \le \frac{\lambda L}{v(1-\rho)}. $$

(20)

Using the bound on the cycle time C _k =sN _k +L _k /v≤sN _k +L/v, we have⁶

$$ \mathbb{E}^0 [C] \le \frac{\rho L}{v(1-\rho)} + \frac{L}{v}. $$

(21)

In order to lower bound the expected cycle duration, we lower bound the expected travel distance per cycle. This distance is at least as large as the expected distance between a uniformly distributed point (message location) in the network region and the center of the region less r ^∗. For a square shaped region of area A, this distance can be lower bounded by $\underline{d} \doteq 0.383\sqrt{A} - r^{*}$ [7]. Therefore, we have

$$ \mathbb{E} [N_{k+1}| N_k] \ge \lambda ( \underline{d}/v + sN_k ). $$

Upon taking expectations, we have

$$ \mathbb{E}^0 \bigl[N^c\bigr] \ge \frac{\lambda \underline{d}}{v(1-\rho)}, $$

(22)

and

$$ \mathbb{E}^0 [C] \ge \frac{\underline{d}}{v(1-\rho)}. $$

(23)

Next, we characterize the second moment of the cycle duration. Let T _s denote the time it takes to serve Poisson arrivals arriving in a time interval of random duration D. If the interarrival times, service times s, and the duration of time D are independent, the second moment of T _s is given by [32, p. 238] or [2, p. 1107]

$$ \mathbb{E} \bigl[T_s^2\bigr] = \lambda^2\mathbb{E}[s]^2\mathbb{E}\bigl[D^2 \bigr] + \lambda\mathbb{E}\bigl[s^2\bigr]\mathbb{E}[D]. $$

This result can be applied to the workload in our system, where the workload for the (k+1)th cycle in our system is given by sN _k+1, and the random duration of interest is the kth cycle duration C _k . The reason we can use the result from [32] in our system is that the duration of the kth cycle is a function of the arrivals in the previous cycle, and it is independent of the interarrival times during the kth cycle. Therefore,

$$ \mathbb{E} \bigl[s^2N_{k+1}^2\bigr] = \lambda^2s^2\mathbb{E}\bigl[C_k^2 \bigr] + \lambda s^2\mathbb{E}[C_k]. $$

(24)

Thus, we have

which upon utilizing the upper bounds on the first moments of N ^c and C in (20) and (21) gives

where we let $\overline{N^{c}}$ denote the finite constant on the right-hand side. Using the bound on $\mathbb{E}^{0}[(N^{c})^{2}]$, we can upper bound the second moment of the cycle duration easily as follows:

(25)

Expected waiting time

Next, we bound the expected waiting time in order to bound the expected number of messages in the system via Little’s law. For this, we first establish that the delay process {W _j :j∈ℤ₊}, and the queue length process {N(t):t≥0} converge to stationary processes.

Lemma 3

The processes {W _j :j∈ℤ₊} and {N(t):t≥0} form positive recurrent regenerative processes under the TSPN policy.

The proof is given at the end of the proof of Theorem 1. It establishes that the times when an arrival finds an empty system with the collector at the center of the network region constitute regeneration epochs for the system. Because the regeneration processes {N(t):t≥0} and {W _j :j∈ℤ₊} are positive recurrent and their regeneration periods are aperiodic, the sequences of message delays converge in distribution to a (customer)-stationary process, denoted by $\tilde{W}$, and the queue length process {N(t):t≥0} converge in distribution to a time-stationary process, denoted by $\tilde{N}$, see [49]. Now, we derive a bound on the expected waiting time, $\mathbb{E}[\tilde{W}]$, according to the stationary delay distribution. This bound is derived in a similar way to [2] or [13]. The delay of an arbitrary message is upper bounded by the sum of the residual cycle time C _R , plus the duration of the next cycle C _N . Note that the cycle during which the arrival occurs is a-typical and has expected duration $\mathbb{E}[C_{R}] + \mathbb{E}[C_{P}]$, where $\mathbb{E}[C_{R}]$ and $\mathbb{E}[C_{P}]$ denote the expected residual and past cycle times and are given by $\mathbb{E}^{0}[C^{2}]/2\mathbb{E}^{0}[C]$ [2, 13]. Therefore,

$$ \mathbb{E}[\tilde{W}] \le \mathbb{E}[C_R] + \mathbb{E}[C_N] \le \frac{\mathbb{E}^0[C^2]}{2\mathbb{E}^0[C]} + \mathbb{E}[C_N]. $$

(26)

Note that C _N is also a-typical and equal to the sum of the travel time plus the amount of workload that arrived during the previous cycle. Therefore, we have [2],

$$ \mathbb{E}[C_N] \le \rho \bigl( \mathbb{E}[C_P] + \mathbb{E}[C_R]\bigr) + \frac{L}{v}. $$

(27)

Finally, combining (27) with the expression for the expected residual time, $\mathbb{E}[C_{R}] = \mathbb{E}^{0}[C^{2}]/2\mathbb{E}^{0}[C]$, we have from (26),

$$ \mathbb{E}[\tilde{W}] \le \biggl(\rho + \frac{1}{2}\biggr) \frac{\mathbb{E}^0[C^2]}{2\mathbb{E}^0[C]} + \frac{L}{v} < \infty $$

where the last inequality holds due to (23) and (25),

Finally, the stationary version of Little’s law gives a relationship between the first moment of the time-stationary process $\tilde{N}$, and the first moment of the customer-stationary process $\tilde{W}$ [55]. We have

$$ \mathbb{E}[\tilde{N}] = \lambda \mathbb{E}[\tilde{W}] < \infty. $$

This establishes the stability of the TSPN policy for any load ρ<1.

Proof of Lemma 3

Let the arrival time of the jth message be $\tilde{t}_{j}$, and its delay W _j . We consider the Markov chain {N _k :k∈ℕ} at the beginning of cycles which is positive recurrent, and therefore, hits the empty state infinitely often. Consecutive epochs and times at which an arrival finds the collector at the center of an empty system (i.e., start of a cycle) constitute an embedded renewal process for both processes {W _j :j∈ℤ₊}, and {N(t):t≥0}. Namely, let the sequence {ℓ _n :n∈ℤ₊} denote the sequence of arrivals that find an empty system with the collector at the center. Because the arrival and the service processes are stationary, the discrete sequence {ℓ _n :n∈ℤ₊} serve as an embedded renewal process for the delay process {W _n :n∈ℤ₊}, and the continuous times $\tilde{t}_{\ell_{n}}$ serve as one for the queue length process {N(t):t≥0}. More precisely, we have that the process is independent of and of $\tilde{t}_{\ell_{1}}$, and the process is stochastically identical to {N(t):t≥0}. Similarly, the process is independent of {W _n :n<ℓ ₁} and of ℓ ₁, and the process is stochastically identical to {W _n :n∈ℤ₊}; see [49].

Next, we show that these renewal processes are positive recurrent. Namely, we show that the expectation of the interrenewal periods, , are finite. Let T _r be the duration of the rth renewal period, where the sequence {T _r :r∈ℤ₊} is i.i.d., and we need to show that $\mathbb{E}[T_{1}]<\infty$. Let m ₀ be the mean recurrence time of the empty state (i.e., the state N _k =0) in the Markov chain {N _k }, which is finite since the Markov chain is positive recurrent. Note that m ₀ also denotes the expected number of cycles between renewals. Given K let M(K) be the number of renewals that have taken place up to and including cycle K. Since the last renewal might have taken place before cycle K, we have

$$ \frac{\sum_{k=1}^K C_k}{\sum_{r=1}^{M(K)}T_r } \ge 1. $$

(28)

Furthermore, we have from Strong Law of Large Numbers (SLLN)

$$ \lim_{K\rightarrow \infty} \frac{M(K)}{K} = \frac{1}{m_0}, \quad \mbox{a.s.} $$

(29)

The extended version of the Strong Law of Large Numbers (SLLN) for nonnegative valued random variables states that if the expectation of the random variables involved is infinite, then their average converges to infinity; see, for example [45, p. 370]. Now, applying the extended version of the SLLN to T _r we have

$$ \lim_{K\rightarrow \infty} \frac{1}{M(K)} \sum _{r=1}^{M(K)}T_r = \mathbb{E}[T_1], \quad \mbox{a.s.} $$

(30)

Note that we will establish that the above expectation is indeed finite. We utilize the upper bound on the cycle times C _k ≤sN _k +L/v in (28) to have

$$ \frac{\sum_{k=1}^K (sN_k+\frac{L}{v} )}{\sum_{r=1}^{M(K)}T_r } \ge 1. $$

(31)

Since the Markov chain {N _k :k∈ℕ} is ergodic, we have

$$ \lim_{K\rightarrow \infty} \frac{1}{K} \sum _{k=1}^{K}N_k = \mathbb{E}^0 \bigl[N^c\bigr],\quad \mbox{a.s.} $$

(32)

Finally, rewriting (31), taking the limit as K tends to infinity, and applying (29), (30), and (32), we have

$$ \lim_{K\rightarrow \infty} \frac{K}{M(K)} \frac{\frac{1}{K}\sum_{k=1}^K sN_k + \frac{L}{v}}{\frac{1}{M(K)}\sum_{r=1}^{M(K)}T_r } = m_0 \frac{s\mathbb{E}^0[N^c] + \frac{L}{v}}{\mathbb{E}[T_1]} \ge 1, $$

which implies that

$$ \mathbb{E}[T_1] \le m_0 \biggl( s\mathbb{E}^0 \bigl[N^c\bigr] + \frac{L}{v} \biggr) = m_0 \biggl( \frac{\rho L}{v(1-\rho)} + \frac{L}{v} \biggr) < \infty, $$

where we used (20) for the last inequality. This establishes the fact that the regenerative processes {W _j :j∈ℤ₊} and {N(t):t≥0} are positive recurrent. □

Appendix B: Proof of Theorem 3

We prove Theorem 3 for a broader class of arrival processes. We assume that each cell i has an arrival process A _i (t) that is i.i.d. over time and satisfies $\mathbb{E}[A_{i}(t)^{2}] \le A_{\max}^{2}$ independent of the number of messages in the system, which is satisfied if the overall arrival process into the system is Poisson. Note that we have $\mathbb{E}[A_{i}(t)] = \lambda_{i} s$ independent of the number of messages in the system. Let t _k , k=0,1,…, be the first time slot of the kth frame. Let D _i (t), t∈{t _k +T _r ,t _k+1−1}, be 1 if cell i is scheduled to be active during the kth frame and zero otherwise. Note that D _i (t) is the service opportunity given to cell i at time slot t and not the actual departure process. Let N _i (t) be the number of messages in cell i at the beginning of time slot t. We assume that arrivals take place at the end of time slots. We have the following queue evolution relation:

Similarly, the following T-step queue evolution expression holds:

The inequality is due to the fact that cell i might become empty and that some arrivals depart during the frame. Squaring both sides we have

(33)

Define the quadratic Lyapunov function

$$ L\bigl(\mathbf{N}(t_k)\bigr) = \sum_{i=1}^K N_i^2(t_k), $$

and the T-step conditional Lyapunov drift

$$ \Delta_T(t_k)\triangleq \mathbb{E} \bigl\{ L\bigl( \mathbf{N}(t_k+T)\bigr) - L\bigl(\mathbf{N}(t_k)\bigr) | \mathbf{N}(t_k) \bigr\}. $$

Summing (33) over the queues, taking conditional expectation, using D _i (t)≤1 for all time slots t, $\mathbb{E}\{A_{i}(t)^{2}\} \le A_{\max}^{2}$ and $\mathbb{E}\{A_{i}(t_{1}\!)A_{i}(t_{2}\!)\} \le \!\! \sqrt{\mathbb{E}\{A_{i}(t_{1}\!)\}^{2}\mathbb{E}\{A_{i}(t_{2}\!)\}^{2}}$ $\le A_{\max}^{2}$ for all t ₁ and t ₂ we have

where $B=1+A_{\max}^{2}$ is a constant. Note that D _i (t+τ)=0,∀i∈{1,…,K} for τ∈{0,1,…,T _r −1} since the system is idle for the first T _r slots of the frame under the FMW policy. Therefore,

Now using the fact that for any load vector ρ=λ s that is strictly inside Λ ⁰, there exist real numbers $\alpha_{1},\ldots,\alpha_{|\mathcal{I}|}$ such that $\alpha_{j} > 0,\forall j \in {1,\ldots,|\mathcal{I}|}$, $\sum_{j=1}^{|\mathcal{I}|}\alpha_{j} = 1-\epsilon$ for some ϵ>0 and

$$ \displaystyle \boldsymbol{\rho}= \sum_{j=1}^{|\mathcal{I}|} \alpha_j \mathbf{I}^j, $$

where I ^j is a K-dimensional vector in $\mathcal{I}$. Over the time interval [t+T _r ,t+T−1], the FMW policy applies the activation vector that has the property

$$ \mathbf{I}^*(t_k) = \displaystyle \arg\max_{\mathbf{I} \in \mathcal{I}} \mathbf{N}(t_k)\cdot\mathbf{I}. $$

(34)

Therefore, ∑ _i N _i (t _k )D _i (t _k +τ)=N(t _k )⋅I ^∗(t _k ). Hence, we have

(35)

Note that we have $\mathbf{N}(t_{k}).\mathbf{I}^{*}(t_{k}) \ge \frac{1}{K}\sum_{i}N_{i}(t_{k})$ since the maximum weight schedule has more weight than the average. Therefore, for $T > \frac{T_{r}}{\epsilon}$ we have

(36)

Therefore, the T-step conditional Lyapunov drift is negative if $T > \frac{T_{r}}{\epsilon}$ and if the queue sizes are outside a bounded set. Therefore, the stability at the frame boundaries follows from Lemma 4.2 in [2] due to a similar reasoning to the proof of Theorem 1. This implies the stability of the system since the frame length T is a constant.