On a non-increasing Lindley-type equation Maria Vlasiou EURANDOM, - PowerPoint PPT Presentation

On a non-increasing Lindley-type equation Maria Vlasiou EURANDOM, Eindhoven email: vlasiou@eurandom.tue.nl CWI Queueing Colloquium. May 27, 2005 1/22 1/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮

Contents 1. The model 2. Stability 3. Successive iterations 4. Derivation of the integral equation 5. The class of separable kernels 6. The distribution of W 7. Tail behaviour 2/22 2/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ Contents

1. The model Service Service • Only one customer point 1 point 2 ���� ���� ���� ���� allowed in a service ���� ���� ���� ���� Service phase point. ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� • Server is necessary ���� ���� ���� ���� Preparation phase ���� ���� ���� ���� ���� ���� ���� ���� only during service ���� ���� ���� ���� phase. � � � � � � � � � � � � • He alternates be- � � � � � � � � tween the two � � � � � � � � service points. � � � � Infinite supply . . . . . . 3/22 3/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } Lindley’s equation for the waiting time in a G/G/ 1 queue: 1 B n (service) W n n -th arrival 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } Lindley’s equation for the waiting time in a G/G/ 1 queue: 1 ��� ��� ��� ��� W n B n (service) W n +1 ��� ��� ��� ��� n -th n + 1-st arrival arrival 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } Lindley’s equation for the waiting time in a G/G/ 1 queue: 1 ��� ��� ��� ��� W n B n (service) W n +1 ��� ��� ��� ��� ��� ��� ��� ��� W n +1 = max { 0 , B n + W n − A n } A n n -th n + 1-st arrival arrival 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

B n : preparation time of the n -th customer A n : service time of the n -th customer W n : waiting time of the server for the n -th customer W n +1 = max { 0 , B n +1 − A n − W n } Lindley’s equation for the waiting time in a G/G/ 1 queue: 1 ��� ��� ��� ��� W n B n (service) W n +1 ��� ��� ��� ��� ��� ��� ��� ��� W n +1 = max { 0 , B n + W n − A n } A n n -th n + 1-st arrival arrival W = max { 0 , B − A − W } In equilibrium: For n � 1 , X n +1 = B n +1 − A n P [ X n < 0] > 0 . and 4/22 4/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

Cases that have been studied: • A deterministic or exponential, B uniform; Park et al. (2003) 5/22 5/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

Cases that have been studied: • A deterministic or exponential, B uniform; Park et al. (2003) • A phase type, B uniform; Vlasiou et al. (2004) 5/22 5/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

Cases that have been studied: • A deterministic or exponential, B uniform; Park et al. (2003) • A phase type, B uniform; Vlasiou et al. (2004) • A general, B phase type; Vlasiou et al. (2005) Here we shall study the " M/G " case; i.e. A exponential, B general. 1 ✄ 7 5/22 5/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ The model

2. Stability • The existence of an invariant distribution is a consequence of the fact that the sequence P [ W n � x ] is tight and the function g ( w, x ) = max { 0 , x − w } is continuous in both x and w ; see Foss and Konstan- topoulos (2004), Theorem 4. 6/22 6/22 Stability ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮

2. Stability • The existence of an invariant distribution is a consequence of the fact that the sequence P [ W n � x ] is tight and the function g ( w, x ) = max { 0 , x − w } is continuous in both x and w ; see Foss and Konstan- topoulos (2004), Theorem 4. • The uniqueness of the steady-state distribution and the convergence to it can be shown by a simple coupling argument. 6/22 6/22 Stability ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮

2. Stability • The existence of an invariant distribution is a consequence of the fact that the sequence P [ W n � x ] is tight and the function g ( w, x ) = max { 0 , x − w } is continuous in both x and w ; see Foss and Konstan- topoulos (2004), Theorem 4. • The uniqueness of the steady-state distribution and the convergence to it can be shown by a simple coupling argument. • Through coupling we can show that P [ X � 0] is a bound of the rate of convergence to the invariant distribution. 6/22 6/22 Stability ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮

3. Successive iterations We have F W ( x ) = P [ W � x ] = P [ X − W � x ] = 1 − P [ X − W � x ] 7/22 7/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ Successive iterations

3. Successive iterations We have F W ( x ) = P [ W � x ] = P [ X − W � x ] = 1 − P [ X − W � x ] � ∞ � ∞ = 1 − P [ W � y − x ] dF X ( y ) = 1 − F W ( y − x ) dF X ( y ) . x x 7/22 7/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ Successive iterations

3. Successive iterations We have F W ( x ) = P [ W � x ] = P [ X − W � x ] = 1 − P [ X − W � x ] � ∞ � ∞ = 1 − P [ W � y − x ] dF X ( y ) = 1 − F W ( y − x ) dF X ( y ) . x x Theorem 1. There is a unique measurable bounded function F : [0 , ∞ ) → R that satisfies the functional equation � ∞ F ( x ) = 1 − F ( y − x ) dF X ( y ) . x 7/22 7/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ Successive iterations

Proof. In L ∞ ([0 , ∞ )) we define the mapping � ∞ ( T F )( x ) = 1 − F ( y − x ) dF X ( y ) . x Then we have � ( T F 1 ) − ( T F 2 ) � = sup x � 0 | ( T F 1 )( x ) − ( T F 2 )( x ) | � ∞ � � � � = sup [ F 2 ( y − x ) − F 1 ( y − x )] dF X ( y ) � � x � 0 � � x � ∞ � sup sup t � 0 | F 2 ( t ) − F 1 ( t ) | dF X ( y ) x � 0 x = � F 1 − F 2 � sup x � 0 (1 − F X ( x )) � � F 1 − F 2 � (1 − F X (0)) = � F 1 − F 2 � P ( B > A ) . 8/22 8/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ Successive iterations

• The convergence to the invariant distribution is geometrically fast and the rate is bounded by the probability P ( B > A ) . 9/22 9/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ Successive iterations

• The convergence to the invariant distribution is geometrically fast and the rate is bounded by the probability P ( B > A ) . • Since we have a contraction mapping, we can approximate the limiting distribution by successive iterations. 9/22 9/22 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮ Successive iterations