在Akka中，可以使用持久化来确保路由器消息的持久性。下面是一个包含代码示例的解决方法：

1. 首先，需要将Akka持久化模块添加到项目的依赖中。在sbt中，可以使用以下命令添加依赖：

   libraryDependencies += "com.typesafe.akka" %% "akka-persistence" % "2.6.16"
   

2. 创建一个持久化Actor来处理路由器消息。这个Actor需要扩展PersistentActor类，并实现receiveCommand和receiveRecover方法。

   import akka.actor.{ActorLogging, Props}
   import akka.persistence.PersistentActor
   
   // 持久化Actor
   class RouterPersistenceActor extends PersistentActor with ActorLogging {
     
     override def persistenceId: String = "router-persistence-actor"
   
     override def receiveCommand: Receive = {
       case msg: Any =>
         // 处理消息
         log.info("Received message: {}", msg)
         // 持久化消息
         persist(msg) { _ =>
           log.info("Message persisted: {}", msg)
         }
     }
   
     override def receiveRecover: Receive = {
       case msg: Any =>
         // 恢复持久化的消息
         log.info("Recovered message: {}", msg)
     }
   }
   
   // 创建持久化Actor的Props
   object RouterPersistenceActor {
     def props: Props = Props[RouterPersistenceActor]
   }
   

3. 创建一个路由器Actor，在需要持久化消息的地方将消息发送给持久化Actor。

   import akka.actor.{Actor, ActorRef, Props}
   import akka.routing.{ActorRefRoutee, RoundRobinRoutingLogic, Router}
   
   // 路由器Actor
   class RouterActor extends Actor {
     private var router: Router = {
       val routees = Vector.fill(5) {
         val r = context.actorOf(RouterPersistenceActor.props)
         context.watch(r)
         ActorRefRoutee(r)
       }
       Router(RoundRobinRoutingLogic(), routees)
     }
   
     override def receive: Receive = {
       case msg: Any =>
         // 将消息发送给持久化Actor
         router.route(msg, sender())
     }
   }
   
   // 创建路由器Actor的Props
   object RouterActor {
     def props: Props = Props[RouterActor]
   }
   

4. 创建一个使用路由器Actor的示例，并发送一些消息。

   import akka.actor.ActorSystem
   
   object ExampleApp extends App {
     val system = ActorSystem("example")
   
     val router = system.actorOf(RouterActor.props)
   
     // 发送一些消息
     router ! "Message 1"
     router ! "Message 2"
     router ! "Message 3"
   
     system.terminate()
   }
   

在上面的示例中，持久化Actor将接收到的消息写入持久化存储。当系统重新启动时，持久化Actor将从持久化存储中恢复消息，并处理它们。这样可以确保消息的持久性。