# 9 Examples of monotonicity

## 9.1 Power-delay tradeoff in wireless communication

In a cell phone, higher layer applications such as voicecall, email, browsers, etc. generate data packets. These packets are buffered in a queue and the transmission protocol decides how many packets to transmit at each time depending the number of packets in the queue and the quality of the wireless channel.

Let \(X_t \in \integers_{\ge 0}\) denote the number of packets buffered at time \(t\) and \(A_t \in \integers_{\ge 0}\), \(A_t \le X_t\), denote the number of packets transmitted at time \(t\). The remaining \(X_t - A_t\) packets incur a delay penalty given by \(d(X_t - A_t)\), where \(d(\cdot)\) is a *strictly* increasing and discrete-convex function where \(d(0) = 0\).

A function \(f \colon \integers \to \reals\) is called convex (or \(L^{\#}\) convex) if for any \(x \in \integers\), \[ f(x+1) + f(x-1) \ge 2 f(x), \] or, equivalently, for any \(x, y \in \integers\) \[ f(x) + f(y) \ge f\Bigl(\Bigl\lfloor \frac{x+y}{2} \Bigr\rfloor\Bigr) + f\Bigl(\Bigl\lceil \frac{x+y}{2} \Bigr\rceil\Bigr).\]

It can be easily seen that \(L^{\#}\) functions satisfy the following properties:

- Sum of \(L^{\#}\) convex functions is \(L^{\#}\) convex.
- Pointwise limits of \(L^{\#}\) convex functions is \(L^{\#}\) convex.

During time \(t\), \(W_t \in \integers_{\ge 0}\) additional packets arrive and \[ X_{t+1} = X_t - A_t + W_t.\] We assume that \(\{W_t\}_{t \ge 1}\) is an i.i.d. process.

The packets are transmitted over a wireless fading channel. Let \(S_t \in \ALPHABET S\) denote the state of the fading channel. We assume that the states are ordered such that a lower value of state denotes a better channel quality.

If the channel has two states, say GOOD and BAD, we typically expect that \[ \PR(\text{GOOD} \mid \text{GOOD}) \ge \PR(\text{GOOD} \mid \text{BAD}). \] This means that the two state transition matrix is stochastically monotone. So, in general (i.e., when the channel has more than two states), we assume that \(\{S_t\}_{t \ge 1}\) is a stochastically monotone Markov process that is independent of \(\{W_t\}_{t \ge 1}\).

The transmission protocol sets the transmit power such that the signal to noise ratio (SNR) at the receiver is above a desired threshold. It can be shown that for additive white Gaussian channels (AWGN), the transmitted power is of the form \[p(A_t) q(S_t),\] where

- \(p(\cdot)\) is a strictly increasing and convex function where \(p(0) = 0\);
- \(q(\cdot)\) is a strictly increasing function.

The objective is to choose a transmission policy \(A_t = π^*_t(X_t, S_t)\) to minimize the weighted sum of transmitted power and delay \[ \EXP\bigg[ \sum_{t=1}^T \big[ p(A_t) q(S_t) + \lambda d(X_t - A_t) \big] \bigg],\] where \(\lambda\) may be viewed as a Lagrange multiplier corresponding to a constrained optimization problem.

### 9.1.1 Dynamic program

We can assume \(Y_t = X_t - A_t\) as a post-decision state in the above model and write the dynamic program as follows:

\[ V^*_{T+1}(x,s) = 0 \] and for \(t \in \{T, \dots, 1\}\), \[\begin{align*} H_t(y,s) &= \lambda d(y) + \EXP[ V^*_{t+1}(y + W_t, S_{t+1}) | S_t = s ], \\ V^*_t(x,s) &= \min_{0 \le a \le x} \big\{ p(a) q(s) + H_t(x-a, s) \big\} \end{align*}\]

### 9.1.2 Monotonicity of value functions

**Lemma 9.1** For all \(t\), \(V^*_t(x,s)\) and \(H_t(y,s)\) are increasing in both variables.

First note that the constraint set \(\ALPHABET A(x) = \{0, \dots, x\}\) satisfies the conditions that generalize the result of monotonicity to constrained actions.

We prove the two monotonicity properties by backward induction. First note that \(V^*_{T+1}(x,s)\) is trivially monotone. This forms the basis of induction. Now suppose \(V^*_{t+1}(x,s)\) is increasing in \(x\) and \(s\). Since \(\{S_t\}_{t \ge 1}\) is stochastically monotone, \[H_t(y,s) = \lambda d(y) + \EXP[ V^*_{t+1}(y + W_t, S_{t+1}) | S_t = s ]\] is increasing in \(s\). Moreover, since both \(d(y)\) and \(V^*_{t+1}(y + w, s)\) are increasing in \(y\), so is \(H_t(y,s)\).

Now, for every \(a\), \(p(a) q(s)\) and \(H_t(x-a, s)\) is increasing in \(x\) and \(s\). So, the pointwise minima over \(a\) is also increasing in \(x\) and \(s\).

### 9.1.3 Convexity of value functions

**Lemma 9.2** For all time \(t\) and channel state \(s\), \(V^*_t(x,s)\) and \(H_t(y,s)\) are convex in the first variable.

We proceed by backward induction. First note that \(V^*_{T+1}(x,s)\) is trivially convex in \(x\). Now assume that \(V^*_{t+1}(x,s)\) is convex in \(x\). Then, \(\EXP[V^*_{t+1}(y + W_t, S_{t+1}) | S_t = s]\) is weighted sum of convex functions and is, therefore, convex in \(y\). Therefore, \(H_t(y,s)\) is a sum of two convex functions and, therefore, convex in \(y\).

We cannot directly show the convexity of \(V^*_t(x,s)\) because the pointwise minimum of convex functions is not convex. So, we consider the following argument. Fix \(s\) and pick \(x > 1\). Let \(\underline a = π^*_t(x-1,s)\) and \(\bar a = π^*_t(x+1,s)\). Let \(\underline v = \lfloor (\underline a + \bar a)/2 \rfloor\) and \(\bar v = \lceil (\underline a + \bar a)/2 \rceil\). Note that both \(\underline v\) and \(\bar v\) are feasible at \(x\). Then, \[ \begin{align*} \hskip 2em & \hskip -2em V^*_t(x-1, s) + V^*_t(x+1, s) \\ &= [ p(\underline a) + p(\bar a) ] q(s) + H_t(x - 1 - \underline a, s) + H_t(x + 1 - \bar a, s) \\ &\stackrel{(a)}\ge [ p(\underline v) + p(\bar v)] q(s) + H_t(x - \underline v, s) + H_t(x - \bar v, s) \\ &\ge 2 \min_{a \le x} \big\{ p(a) q(s) + H_t(x-a, s) \\ &= 2 V^*_t(x,s), \end{align*} \] where \((a)\) follows from convexity of \(p(\cdot)\) and \(H_t(\cdot, s)\). Thus, \(V^*_t(x,s)\) is convex in \(x\). This completes the induction step.

### 9.1.4 Monotonicity of optimal policy in queue length

**Theorem 9.1** For all time \(t\) and channel state \(s\), there is an optimal strategy \(π^*_t(x,s)\) which is increasing in the queue length \(x\).

In Lemma 9.2, we have shown that \(H_t(y,s)\) is convex in \(y\). Therefore, \(H_t(x-a, s)\) is submodular in \((x,a)\).

Thus, for a fixed \(s\), \(p(a)q(s) + H_t(x-a, s)\) is submodular in \((x,a)\). Therefore, the optimal policy is increasing in \(x\).

### 9.1.5 Lack of monotonicity of optimal policy in channel state

It is natural to expect that for a fixed \(x\) the optimal policy is decreasing in \(s\). However, it is not possible to obtain the monotonicity of optimal policy in channel state in general. To see why this is difficult, let us impose a mild assumption on the arrival distribution.

#### (**asm-power-delay-density?**)

The packet arrival distribution is weakly decreasing, i.e., for any \(v,w \in \integers_{\ge 0}\) such that \(v \le w\), we have that \(P_W(v) \ge P_W(w)\).

We first start with a slight generalization of stochastic monotonicity result.

**Lemma 9.3** Let \(\{p_i\}_{i \ge 0}\) and \(\{q_i\}_{i \ge 0}\) be real-valued non-negative sequences satisfying \[ \sum_{i \le j} p_i \le \sum_{i \le j} q_i, \quad \forall j.\] (Note that the sequences do not need to add to 1). Then, for any increasing sequence \(\{v_i\}_{i \ge 0}\), we have \[ \sum_{i = 0}^\infty p_i v_i \ge \sum_{i=0}^\infty q_i v_i. \]

The proof is similar to the proof for stochastic monotonicity.

**Lemma 9.4** Under (**asm-power-delay-density?**), for all \(t\), \(H_t(y,s)\) is supermodular in \((y,s)\).

The idea of the proof is similar to Lemma 8.1.

Fix \(y^+, y^- \in \integers_{\ge 0}\) and \(s^+, s^- \in \ALPHABET S\) such that \(y^+ > y^-\) and \(s^+ > s^-\). Now, for any \(y' \in \integers_{\ge 0}\) and \(s' \in \ALPHABET S\) define \[\begin{align*} π(y',s') = P_W(y' - y^+)P_S(s'|s^+) + P_W(y' - y^-)P_S(s'|s^-), \\ μ(y',s') = P_W(y' - y^-)P_S(s'|s^+) + P_W(y' - y^+)P_S(s'|s^-). \end{align*}\]

Since \(P_S\) is stochastically monotone, we have that for any \(σ \in \ALPHABET
S\), \[ \sum_{s'=1}^{σ} P_S(s'|s^+) \le \sum_{s'=1}^{σ} P_S(s'|s^-). \] Moreover, due to (**asm-power-delay-density?**), we have that \(P_W(y' - y^-)
\le P_W(y' - y^+)\). Thus, \[ [P_W(y' - y^+) - P_W(y' - y^-)] \sum_{s'=1}^{σ} P_S(s'|s^+)
\le [P_W(y' - y^+) - P_W(y' - y^-)]\sum_{s'=1}^{σ} P_S(s'|s^-). \] Rearranging terms, we get \[ \sum_{s'=1}^σ π(y',s') \le \sum_{s'=1}^σ μ(y',s'). \] Thus, for any \(y'\), the sequence \(π(y',s')\) and \(ν(y',s')\) satisfy the condition of Lemma 9.3.

Now, in Lemma 9.1, we have established that for any \(y'\), \(V_{t+1}(y',s')\) is increasing in \(s'\). Thus, from Lemma 9.3, we have \[ \sum_{s' \in \ALPHABET S} π(y', s') V_{t+1}(y', s') \ge \sum_{s' \in \ALPHABET S} μ(y', s') V_{t+1}(y', s'), \] Summing up over \(y'\), we get \[ \sum_{y' \in \integers_{\ge 0}} \sum_{s' \in \ALPHABET S} π(y', s') V_{t+1}(y', s') \ge \sum_{y' \in \integers_{\ge 0}} \sum_{s' \in \ALPHABET S} μ(y', s') V_{t+1}(y', s'), \] Or equivalently, \[\begin{align*} \hskip 2em & \hskip -2em \EXP[ V_{t+1}(y^+ + W, S_{t+1}) | S_t = s^+) ] + \EXP[ V_{t+1}(y^- + W, S_{t+1}) | S_t = s^-) ] \\ & \ge \EXP[ V_{t+1}(y^- + W, S_{t+1}) | S_t = s^+) ] + \EXP[ V_{t+1}(y^+ + W, S_{t+1}) | S_t = s^-) ] . \end{align*}\] Thus, \(H_t(y,s)\) is supermodular in \((y,s)\).

**asm-power-delay-density?**), we cannot establish the monotonicity of \(π^*_t(x,s)\) is \(s\).

Note that we have established that \(H_t(y,s)\) is supermodular in \((y,s)\). Thus, for any fixed \(x\), \(H_t(x-a,s)\) is submodular in \((a,s)\). Furthermore the function \(p(a)q(s)\) is increasing in both variables and therefore supermodular in \((a,s)\). Therefore, we cannot say anything specific about \(p(a)q(s) + H_t(x-a, s)\) which is a sum of submodular and supermodular functions.

We need to impose a much stronger assumption to establish monotonicity in channel state. See Exercise 9.1.

## Exercises

**Exercise 9.1** Suppose that the channel state \(\{S_t\}_{t \ge 1}\) is an i.i.d. process. Then prove that for all time \(t\) and queue state \(x\), there is an optimal strategy \(π^*_t(x,s)\) which is decreasing in channel state \(s\).

## Notes

The mathematical model of power-delay trade-off is taken from Berry (2000), where the monotonicty results were proved using first principles. More detailed characterization of the optimal transmission strategy when the average power or the average delay goes to zero are provided in Berry and Gallager (2002) and Berry (2013). A related model is presented in Ding et al. (2016). A slight generalization of this model is also considered in Fu and Schaar (2012) where monotonicty in the queue state is stablished.

For a broader overview of power-delay trade offs in wireless communication, see Berry et al. (2012) and Yeh (2012).

The remark after Lemma 9.4 shows the difficulty in establishing monotonicity of optimal policies for a multi-dimensional state space. In fact, sometimes even when monotonicity appears to be intuitively obvious, it may not hold. See Sayedana and Mahajan (2020) for an example. For general discussions on monotonicity for multi-dimensional state spaces, see Topkis (1998) and Koole (2006). As an example of using such general conditions to establish monotonicity, see Sayedana et al. (2020).