Joint routing and scheduling control in a two-class network with a flexible server

Abstract

In this study we analyze a queueing model with a Gurvich structure. In such a network, the controller may route incoming jobs to different classes, but they are routed to the same server. This structure, although it falls into the general class of stochastic processing networks, is somewhat unconventional. We focus on a single-server two-class version of a Gurvich network in this paper. For a Poisson arrival stream and exponential service rates, we develop a Markov decision process representation of the system and prove structural results on optimal routing and scheduling controls. We show that the optimal policy uses $c\mu$ scheduling and switching curve routing. We also investigate the fluid model and perturbation expansions thereof, which are useful in deriving near-optimal policies in the original network.

Appendix

Theorem 3.2 is established via the following five lemmas below.

Lemma 8.1

If Statements 1 and 5 hold for n, then Statement 1 holds for $n+1$.

Proof

Statement 1 for $n+1$ is

$$\begin{aligned} v^{n+1}(i,j, C_1) < v^{n+1}(i,j, C_2) \end{aligned}$$

for all $i, j \ge 1$. By expanding the value functions on both sides, it is clear that both sides have the same single-step costs and terms related to arrivals. Thus, if the following inequality for the departures holds, then the result is established:

$$\begin{aligned}&\mu _1 \min (v^{n}(i-1,j, C_1), v^{n}(i-1,j, C_2)) \\&\quad < \mu _2 \min (v^{n}(i,j-1, C_1),v^{n}(i,j-1, C_2))+ (\mu _1 - \mu _2) v^{n}(i,j, C_2). \end{aligned}$$

Using Statement 1 for n, this reduces to

$$\begin{aligned} \mu _1 v^{n}(i-1,j, C_1) < \mu _2 v^{n}(i,j-1, C_1) + (\mu _1 - \mu _2) v^{n}(i,j, C_2), \end{aligned}$$

which corresponds to Statement 5 at n. Therefore, Statement 1 holds for $n+1$. $\square$

Lemma 8.2

If Statements 1 and 2 hold for n, then Statement 2 holds for $(n+1)$.

Proof

Given $v_{\delta }^{n}(i,j,C_1) \le v_{\delta }^{n}(i,j,C_2)$, then the first half of Statement 2 for $n+1$ can be expressed as follows:

$$\begin{aligned} v_{\delta }^{n+1}(i\!+\!1,j,C_1)\!-\! v_{\delta }^{n+1}(i,j,C_1)= & {} c_1 \!+\! \delta \lambda [\min (v_{\delta }^{n}(i\!+\!2,j,C_1), v_{\delta }^{n}(i\!+\!1,j\!+\!1,C_1)) \\&-\,\min (v_{\delta }^{n}(i+1,j,C_1),v_{\delta }^{n}(i,j+1,C_1))] \\&+\,\delta \mu _1 (v_{\delta }^{n}(i,j,C_1)- v_{\delta }^{n}(i-1,j,C_1)) \\\ge & {} c_1 + \delta \lambda [\min (v_{\delta }^{n}(i+2,j,C_1) \\&-\,v_{\delta }^{n}(i+1,j,C_1) , v_{\delta }^{n}(i+1,j+1,C_1) \\&-\,v_{\delta }^{n}(i,j+1,C_1))] \\&+\,\delta \mu _1(v_{\delta }^{n}(i,j,C_1)- v_{\delta }^{n}(i-1,j,C_1)). \end{aligned}$$

By assumption, $\min (v_{\delta }^{n}(i+2,j,C_1) - v_{\delta }^{n}(i+1,j,C_1) , v_{\delta }^{n}(i+1,j+1,C_1) - v_{\delta }^{n}(i,j+1,C_1)) \ge 0$ and $v_{\delta }^{n}(i,j,C_1) - v_{\delta }^{n}(i-1,j,C_1) \ge 0$. Thus, $v_{\delta }^{n+1}(i+1,j,C_1)- v_{\delta }^{n+1}(i,j,C_1)$ is nonnegative. A symmetric argument can be used to prove the second half of Statement 2. $\square$

Lemma 8.3

If Statements 1 and 3 hold for n, then Statement 3 holds for $(n+1)$.

Proof

$\Delta ^{n+1}(i,j)$ can be written as follows:

$$\begin{aligned} \Delta ^{n+1}(i,j)= & {} (c_1 - c_2) \\&+\,\delta \lambda \big (\min (v_{\delta }^{n}(i+2,j,C_{1}),v_{\delta }^{n}(i+1,j+1,C_{1})) \\&-\,\min (v_{\delta }^{n}(i+2,j,C_{1}),v_{\delta }^{n}(i+1,j+2,C_{1}))\big ) \\&+\, \delta \mu _1(v_{\delta }^{n}(i,j,C_{1})-v_{\delta }^{n}(i-1,j+1,C_{1})). \end{aligned}$$

We will prove that each term on the right-hand side above is non-decreasing in i and non-increasing in j, when the other variable is fixed. The first term is constant and hence plays no role in the claims. The second term can be expressed as follows (omitting $\delta \lambda$ for simplicity):

$$\begin{aligned}&\min (v_{\delta }^{n}(i+2,j,C_{1}),v_{\delta }^{n}(i+1,j+1,C_{1})) \\&\qquad - \min (v_{\delta }^{n}(i+1,j+1,C_{1}), v_{\delta }^{n}(i,j+2,C_{1})) \\&\quad = \min (v_{\delta }^{n}(i+2,j,C_{1})- v_{\delta }^{n}(i+1,j+1,C_{1}),0) \\&\qquad + \max (v_{\delta }^{n}(i+1,j+1,C_{1})- v_{\delta }^{n}(i,j+2,C_{1}),0) \\&\quad = \min (\Delta ^{n}(i+1,j),0) + \max (\Delta ^{n}(i,j+1),0). \end{aligned}$$

By assumption, $\min (\Delta ^{n}(i+1,j),0) + \max (\Delta ^{n}(i,j+1),0)$ is non-decreasing in i and non-increasing in j. As for the last term, it is equal to $\delta \mu _1\Delta ^{n}(i-1,j)$ and is therefore non-decreasing in i and non-increasing in j by assumption. $\square$

For the remaining proofs, the discount factor $\delta$ appears on both sides of the inequalities; therefore, we omit it to simplify the derivations.

Lemma 8.4

If Statements 1, 2, 3 and 4 hold for n, then Statement 4 holds for $n+1$.

Proof

Recall that the value function at $n+1$ of a particular state-pair combination is composed of one-step costs and terms related to arrivals and departures. In order to prove that Statement 4 holds for $n+1$, we decompose the value functions at n into these components. Below, we compare the values of each component individually for the left- and right-hand side of the inequality in Statement 4.

• For the one-step costs we have

  $$\begin{aligned} \mu _1(c_1 (i-1) + c_2 j)\le & {} \mu _2 (c_1 i + c_2 (j-1)) + (\mu _1 - \mu _2)(c_1 i +c_2 j) \nonumber \\ \end{aligned}$$

  (8.1)
  $$\begin{aligned} \iff 0\le & {} c_1 \mu _1 - c_2 \mu _2 . \end{aligned}$$

  (8.2)

  Since (8.2) holds due to the parameter assumptions, so does the relation in (8.1) for the one-step costs.

• Departure costs: By Statement 1 at n, $v_{\delta }^{n}(i,j,C_1) \le v_{\delta }^{n}(i,j,C_2)$ for all ${i}\ge 0$ and ${j}\ge 0$. Thus, a class 1 customer is always processed before a class 2 customer. Therefore the value function terms relating to departures for $n+1$ hold by virtue of the assumptions of the lemma:

  $$\begin{aligned} \mu ^2_1 v_{\delta }^{n}(i-2, j, C_1) \le \mu _1 \mu _2 v_{\delta }^{n} (i-1, j-1, C_1) + \mu _1 (\mu _1 - \mu _2) v_{\delta }^{n}(i-1,j, C_1). \end{aligned}$$

• Arrival costs: The inequality in Statement 4 involves three minima, with two arguments each. Hence, to compare arrival terms, in theory, all eight possible routing combinations need to be considered. However, due to relations implied by the induction assumptions, the number of cases to be examined can be winnowed. Statement 3 gives us the following implications: (a) if the routing action for $v_{\delta }^{n}(i,j-1, C_1)$ is $R1_{NP}$, then the routing action for $v_{\delta }^{n}(i-1,j, C_1)$ is $R1_{NP}$, (b) if the routing action for $v_{\delta }^{n}(i,j-1, C_1)$ is $R1_{NP}$, then the routing action for $v_{\delta }^{n}(i,j, C_1)$ is $R1_{NP}$, (c) if the routing action for $v_{\delta }^{n}(i,j, C_1)$ is $R1_{NP}$, then the routing action for $v_{\delta }^{n}(i-1,j, C_1)$ is $R1_{NP}$.

Table 1 shows the remaining possible combinations. Note that $v_{\delta }^{n}(i,j, C_1) \le v_{\delta }^{n}(i,j, C_2)$ implies that in any state of the form $(i,j,C_1)$, preemption is suboptimal. Such states are therefore omitted.

Table 1 Routing actions for states $(i-1, j,C_1),(i, j-1,C_1),(i, j,C_1)$

Statement 4 for Cases 1 and 2 follows immediately from the induction assumption because the inequality for $n+1$ in these cases is just a restatement of the inequality for n (after multiplying by $\lambda$).

To establish Case 3 we need to show that

$$\begin{aligned} \lambda \mu _1 v_{\delta }^{n}(i, j ,C_1) \le \lambda \mu _2 v_{\delta }^{n}(i, j, C_1) + \lambda (\mu _1 - \mu _2) v_{\delta }^{n}(i+1, j, C_1). \end{aligned}$$

This can be rewritten as

$$\begin{aligned} (\mu _1-\mu _2) v_{\delta }^{n}(i, j ,C_1) \le (\mu _1 - \mu _2) v_{\delta }^{n}(i+1, j, C_1), \end{aligned}$$

which holds by invoking Statement 2.

Next, to establish Case 4 we need to show that

$$\begin{aligned} \lambda \mu _1 v_{\delta }^{n}(i, j ,C_1) \le \lambda \mu _2 v_{\delta }^{n}(i, j, C_1) + \lambda (\mu _1 - \mu _2) v_{\delta }^{n}(i, j+1, C_1). \end{aligned}$$

This can be rewritten as

$$\begin{aligned} (\mu _1-\mu _2) v_{\delta }^{n}(i, j ,C_1) \le (\mu _1 - \mu _2) v_{\delta }^{n}(i, j+1, C_1), \end{aligned}$$

which again holds by invoking Statement 2. $\square$

Lemma 8.5

If Statements 1, 2, 3, 4, and 5 hold for n, then Statement 5 holds for $n+1$.

Proof

Notice that the only difference in the Statement 4 and Statement 5 inequalities is the last argument of the last terms, i.e., there is a $C_2$ instead of a $C_1$. Thus, Statement 5 follows immediately from Statement 1 and Statement 4, already proven for the $n+1$ case. $\square$