Yii urlManager允许使用空参数的RESTful规则

I currently have a rule that allows for querying a couple different models according to name, brandId, and ingredients. This rule works, except when I try to pass in an empty value for any of the optional parameters. I thought that I used the correct RegEx but apparently not. I've tried both:

array('ws/service/query', 'pattern'=>'ws/service/query/<model:\w+>/(<name:\w+>)?/(<brand:\d+>)?/(<ingredients:.+>)?', 'verb'=>'GET'),

and

array('ws/service/query', 'pattern'=>'ws/service/query/<model:\w+>/<name:(\w+)?>/<brand:(\d+)?>/<ingredients:(.+)?>', 'verb'=>'GET'),

to no avail. When I type the query string: /ws/service/query/myModel/// I get PHP Error - Undefined Index 'Model'. But when I use a query string like /ws/service/query/myModel/a/1/a I get the desired result.

How do I allow the URL to accept null values for GET parameters?

Here is the action:

public function actionQuery()
    {
                $this->_checkAuth();

        switch ($_GET['model'])
                {
                        case 'dogFood':
                                $criteria = new CDbCriteria();
                                if ($_GET['name']) {
                                        $criteria->addSearchCondition('name_df', $_GET['name']);
                                }
                                if ($_GET['ingredients']) {
                                        $ingredientsArray = explode(',',$_GET['ingredients']);
                                        foreach ($ingredientsArray as $ingredient) {
                                                $criteria->addSearchCondition('ingredients_df', $ingredient);
                                        }
                                }
                                if ($_GET['brand']) {
                                        $criteria->addColumnCondition(array('brand_df' => $_GET['brand']));
                                }
                                $models = DogfoodDf::model()->findAll($criteria);
                                break;
                        default:
                                $this->_sendResponse(501, sprintf(
                                        'Error: Mode <b>query</b> is not implemented for model <b>%s</b>',
                                        $_GET['model']));
                                exit;
                }

                if (empty($models)) {
                        $this->_sendResponse(200,
                                sprintf('No items were found for model <b>%s</b>', $_GET['model']));
                }
                else {
                        $rows = array();

                        foreach ($models as $model) {
                                $rows[] = $model->attributes;
                        }

                        $this->_sendResponse(200, CJSON::encode($rows));
                }
    }