unsigned long period1StartMillis; //start counting period1 time
unsigned long period2StartMillis; //start counting period2 time
unsigned long currentMillis;
const unsigned long fanPeriod1 = 2000; //fan duration
const unsigned long fanPeriod2 = 4000; //fan duration
const int BUTTON1_PIN = A1; // Arduino pin connected to button's pin
const int BUTTON2_PIN = A2; // Arduino pin connected to button's pin
const int LED1_PIN    = 2; // Arduino pin connected to LED's pin
const int LED2_PIN    = 3; // Arduino pin connected to LED's pin
const int ANALOG_PIN  = 5; // Arduino pin connected to analog output pin

// variables will change:
int led1State = 0;     // the current state of LED1
int led2State = 0;     // the current state of LED2
int lastButton1State;    // the previous state of button
int currentButton1State; // the current state of button
int lastButton2State;    // the previous state of button
int currentButton2State; // the current state of button

void setup() {
  pinMode(BUTTON1_PIN, INPUT_PULLUP); // set arduino pin to input pull-up mode
  pinMode(BUTTON2_PIN, INPUT_PULLUP); // set arduino pin to input pull-up mode
  pinMode(LED1_PIN, OUTPUT);          // set arduino pin to output mode
  pinMode(LED2_PIN, OUTPUT);          // set arduino pin to output mode
  digitalWrite(LED1_PIN, led1State);
  digitalWrite(LED2_PIN, led2State);

}

void loop()
{
  currentMillis = millis();
  lastButton1State    = currentButton1State;      // save the last state
  lastButton2State    = currentButton2State;      // save the last state

  //Period1 (duration1):

  currentButton1State = digitalRead(BUTTON1_PIN); // read new state Button #1
  if (lastButton1State == 1 && currentButton1State == 0)
  {
    digitalWrite(LED1_PIN, 1); // LED#1 turn on
    digitalWrite(LED2_PIN, 0);
    analogWrite(ANALOG_PIN, 128); // set analog to 50%
    period1StartMillis = currentMillis;  //save the time that the state change occured
    period2StartMillis = 0;               // turn period 2 off
  }
  if (period1StartMillis && currentMillis - period1StartMillis >= fanPeriod1) // check if time has run out
  {
    digitalWrite(LED1_PIN, 0);
    analogWrite(ANALOG_PIN, 0); // set analog to 0%  This is the PROBLEM #1
    period1StartMillis = 0;
  }

  //Period2 (duration2):

  currentButton2State = digitalRead(BUTTON2_PIN); // read new state Buttom #2
  if (lastButton2State == 1 && currentButton2State == 0)
  {
    digitalWrite(LED2_PIN, 1); //LED#2 turn on
    digitalWrite(LED1_PIN, 0);
    analogWrite(ANALOG_PIN, 255);
    period2StartMillis = currentMillis;  //save the time that the state change occured
    period1StartMillis = 0;               // turn period 1 off
  }
  if (period2StartMillis && currentMillis - period2StartMillis >= fanPeriod2) // check if time has run out
  {
    digitalWrite(LED2_PIN, 0);
    analogWrite(ANALOG_PIN, 0); // set analog to 0%  This is the PROBLEM #2
    period2StartMillis = 0;
  }
}