Analysis of M/M/n/K Queue with Multiple Priorities

Transcription

1 Analysis of M/M/n/K Queue with Multile Priorities Coyright, Sanjay K. Bose For a P-riority system, class P of highest riority Indeendent, Poisson arrival rocesses for each class with i as average arrival rate for class i Service times for each class are indeendent of each other and of the arrival rocesses and are exonentially distributed with mean / i for class i Both Non-reemtive and Preemtive Priority Service discilines are considered Since the service times are exonentially distributed (i.e. memory less), the results for reemtive resume and reemtive non-resume will be identical Coyright, Sanjay K. Bose

2 Solution Aroach Define System State aroriately Draw the corresonding State Transition Diagram with the aroriate flows between the states Write and solve the balance equations to obtain the system state robabilities Note that we have given here the solution aroach that may be taken to solve a queueing roblem of this kind. This has been illustrated with simle examles. More comlex cases may be similarly formulated and solved with a corresonding increase in the solution comlexity Coyright, Sanjay K. Bose 3 M/M/-/- Queue with Preemtive Priority For a P-riority queue of this tye, define the system state as the following P-tule where (n, n,,n P ) n i = Number of jobs of riority class i in the queue i=,..,p Note that the server will always be engaged by a job of the highest riority class resent in the system, i.e. by a job of class j with service rate j if n j and n j+ =...=n P =. Coyright, Sanjay K. Bose 4

3 We illustrate the aroach first for a -riority M/M// queue,,,,,3,,,, 3, Coyright, Sanjay K. Bose 5 The corresonding balance equations for the -riority M/M// queue will be given by (,,, ( ( ( ) = + + +, + ) = + ) = + ) =,,,,,, These may be solved to obtain the desired state robabilities We illustrate next how this aroach may be generalized to aly to queues with finite caacity and/or multile servers, Coyright, Sanjay K. Bose 6 3

4 -Priority M/M//3 Queue (Preemtive Priority) (An examle of a queue with finite caacity),,,,,3,,, 3, New arrivals will be lost if they come when the system is in any of the circled states State Transition Diagram for the -Priority M/M//3 Queue with Preemtive Priority Coyright, Sanjay K. Bose 7 The corresonding balance equations are (,,,,,,,,3 3, ( ( ( ( ( + ) =,,,,, + + ) =,,,,, + + ) =,,, + + ) = + + ) = = = + + ) = = =,,,3,, 3, Normalization Condition,,3, 3,, =,,,, Coyright, Sanjay K. Bose 8 4

5 We can solve for the state robability distribution by solving any nine of the ten balance equation along with the equation for the normalization condition Job loss robability (or the blocking robability) =, +, + 3, +,3 Other desired robabilities may also be found from these state robabilities. Some examles are - P{server busy serving low riority job} =, +, + 3, P{one high riority job in the system} =, +, +, Coyright, Sanjay K. Bose 9 -Priority M/M//3 Queue (Preemtive Priority) (An examle of a queue with finite caacity and multile servers),,,3,,,,, 3, New arrivals will be lost if they come when the system is in any of the circled states Solve in the usual manner for the system state robabilities Coyright, Sanjay K. Bose 5

6 M/M/-/- Queue with Non-reemtive Priority We can roose two different methods of reresenting the system state for a M/M/c/K queue of this tye with P riority classes. Aroach I : If P < c, then this aroach gives a more comact reresentation using a P-tule than the more general Aroach II given next. State Reresentation (n,,n P, s,.,s P ) where n j = number of jobs of class j in system j=,,p s k = number of servers currently busy serving jobs of riority class k k=,..,p Coyright, Sanjay K. Bose Aroach II : This requires a (P+c)-tule of the following form State Reresentation n,,n P, s,.,s c ) where n j = number of jobs of class j in system j=,,p s k = riority class of the service currently on-going at server k k=,..,c Note that n +...+n P K for a finite caacity system We have used the reresentation of Aroach II in the examle described subsequently Coyright, Sanjay K. Bose 6

7 -Priority M/M//3 Queue (Non-reemtive Priority) (An examle of a single server queue with finite caacity),,,,,,,,,,,,,,,,,,,3,,, 3,, State Transition Diagram Coyright, Sanjay K. Bose 3 The balance equations for this queue are (,,,,,,,,,,,, ( ( ( ( + ) = ( ( ,, + ) = + ) = + ) = + ) =,, + ) = + ) =,,,,,,,,,,,,,,,,,,,,,,,3, 3,,.and. Coyright, Sanjay K. Bose 4 7

8 ,,,,,,,,,3, 3,, = = = = = =,,,,,,,,,,,,,,,, with the following normalization condition,,,,,,,,,,,,,3,,,,,,, 3,,,, = Coyright, Sanjay K. Bose 5 These equations may be solved on the usual way to obtain the individual state robabilities as er the definition of the system state These state robabilities may then be used to comute other erformance arameters and robabilities that may be of interest. For examle, the blocking robability of this system will be given by (,3, 3,,,,,,,,,, ) Other, similar robabilities and erformance measures may also be calculated Coyright, Sanjay K. Bose 6 8

9 Aroach may be extended in the usual fashion to analyze other similar systems as follows - More than two riority classes Other buffer caacity values or even queues with infinite buffer caacities Different caacity limits for the different riority classes Queues with more than one server Coyright, Sanjay K. Bose 7 9