Approximate Waiting Time Analysis of Priority Queues
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2 Service Parameters
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The special H and general G service time distributions are easy to compute for the highest priority level. Arriving into an empty queue the highest priority customer spends H time in the system. This consists of the remaining service time of the currently processed lower priority customer Sir if any and its own service time S Figure 3. The general service time G is equivalent with the service time of the original service process, because in priority systems the server does not serve lower priority customers while there is highest priority customer in the system. Consequently:
1 2 3
di erent priority levels are. A priority queue is characterized by the following parameters: for each priority, the arrival parameters i , cAi , for each priority, the service parameters i, cSi . From these parameters, we have to compute the parameters of the independent" G G 1Spec. Service queues associated with the di erent priority levels: the arrival parameters i , cAi , the normal non-empty service parameters E Gi , cGi , the special empty service parameters E Hi, cHi .
one queue. We compute the required parameters of this aggregated queue arrival parameters , cA ; service parameters , cS , then this aggregated queue is considered as the only higher priority queue for the analysis of queue j . This method simpli es the analysis signi cantly compared to separate analysis of higher priority queues. If a customer j arrives into an empty queue, the service time it will see is Hj . The components of this special service time are as follows. If the whole system is empty, the special service time only consists of its own service time Sj . If a higher priority customer is under service, the arrived priority j customer has to wait for the remaining service time of the currently processed customer S r , the time required by the higher priority queues to become empty R N , and its own service time Sj . If a lower priority i customer is under service, the arrived customer has to wait for the remaining service time of the currently processed customer Sir , the time required by the higher priority queues to become empty starting from the number of arrivals occurred during Si R CA Si, and its own service time Si. Note that there is no higher priority customer in the system when the service of priority i customer starts. Figure 2 shows the case when a lower priority customer is under service when a priority j customer arrives. On Figure 1 and Figure 2 dotted lines denote higher priority, dashed lines denote lower priority events, solid lines mean priority j events.
1 1Βιβλιοθήκη 2.1 Highest Priority Queue
H1 =
S1 Sir + S1 G1 = S1 ;
qe 1 qi1
i 2 f2; : : : ; Qg;
1 2
where "R" is the number of priority levels. qij is the probability that there is a priority i customer in the server at the arrival instant of a priority j customer, and at that arrival instant queue j was empty. For qij we apply the following approximation: Each priority i ow brings i work into the system i i in a unit of time, which has to be served if the system is stable, hence the probability of serving a priority j customer in steady state is j priority does not destroy work conservativity. That is:
Approximate Waiting Time Analysis of Priority Queues
G bor Horv th a a Dept. of Telecommunications, Technical University of Budapest hgabor@webspn.hit.bme.hu
Abstract
1 Introduction
In our priority system there are as many queues as many priority classes. When the server becomes idle, it starts serving a customer from the highest priority non-empty queue. To characterize the service an arriving customer of priority r receives, the following cases need to be considered. The service time of the customer arrives into an empty queue" there is no customer of the same priority in the system di ers from the case it arrives into a non-empty queue". If the customer arrives into an empty queue, the components of its system time measured from its arrival are: the remaining service time of the customer currently under service, the busy period of the higher priority queues the time required by the higher priority queues to become empty, the service time of the given customer. If the customer arrives into a non-empty queue, the components of its service time which is measured from the last departure of a customer with the same priority are: the busy period of the higher priority queues becoming the higher priority queues empty, the service time of the given priority r customer. This kind of behavior the service time di ers if the queue is empty is called special service, and the concept is suitable for the analysis of a single priority level in isolation. The priority queue is converted to as many separate G G 1-Special Service queues as many
2 Service Parameters
1
The special H and general G service time distributions are easy to compute for the highest priority level. Arriving into an empty queue the highest priority customer spends H time in the system. This consists of the remaining service time of the currently processed lower priority customer Sir if any and its own service time S Figure 3. The general service time G is equivalent with the service time of the original service process, because in priority systems the server does not serve lower priority customers while there is highest priority customer in the system. Consequently:
1 2 3
di erent priority levels are. A priority queue is characterized by the following parameters: for each priority, the arrival parameters i , cAi , for each priority, the service parameters i, cSi . From these parameters, we have to compute the parameters of the independent" G G 1Spec. Service queues associated with the di erent priority levels: the arrival parameters i , cAi , the normal non-empty service parameters E Gi , cGi , the special empty service parameters E Hi, cHi .
one queue. We compute the required parameters of this aggregated queue arrival parameters , cA ; service parameters , cS , then this aggregated queue is considered as the only higher priority queue for the analysis of queue j . This method simpli es the analysis signi cantly compared to separate analysis of higher priority queues. If a customer j arrives into an empty queue, the service time it will see is Hj . The components of this special service time are as follows. If the whole system is empty, the special service time only consists of its own service time Sj . If a higher priority customer is under service, the arrived priority j customer has to wait for the remaining service time of the currently processed customer S r , the time required by the higher priority queues to become empty R N , and its own service time Sj . If a lower priority i customer is under service, the arrived customer has to wait for the remaining service time of the currently processed customer Sir , the time required by the higher priority queues to become empty starting from the number of arrivals occurred during Si R CA Si, and its own service time Si. Note that there is no higher priority customer in the system when the service of priority i customer starts. Figure 2 shows the case when a lower priority customer is under service when a priority j customer arrives. On Figure 1 and Figure 2 dotted lines denote higher priority, dashed lines denote lower priority events, solid lines mean priority j events.
1 1Βιβλιοθήκη 2.1 Highest Priority Queue
H1 =
S1 Sir + S1 G1 = S1 ;
qe 1 qi1
i 2 f2; : : : ; Qg;
1 2
where "R" is the number of priority levels. qij is the probability that there is a priority i customer in the server at the arrival instant of a priority j customer, and at that arrival instant queue j was empty. For qij we apply the following approximation: Each priority i ow brings i work into the system i i in a unit of time, which has to be served if the system is stable, hence the probability of serving a priority j customer in steady state is j priority does not destroy work conservativity. That is:
Approximate Waiting Time Analysis of Priority Queues
G bor Horv th a a Dept. of Telecommunications, Technical University of Budapest hgabor@webspn.hit.bme.hu
Abstract
1 Introduction
In our priority system there are as many queues as many priority classes. When the server becomes idle, it starts serving a customer from the highest priority non-empty queue. To characterize the service an arriving customer of priority r receives, the following cases need to be considered. The service time of the customer arrives into an empty queue" there is no customer of the same priority in the system di ers from the case it arrives into a non-empty queue". If the customer arrives into an empty queue, the components of its system time measured from its arrival are: the remaining service time of the customer currently under service, the busy period of the higher priority queues the time required by the higher priority queues to become empty, the service time of the given customer. If the customer arrives into a non-empty queue, the components of its service time which is measured from the last departure of a customer with the same priority are: the busy period of the higher priority queues becoming the higher priority queues empty, the service time of the given priority r customer. This kind of behavior the service time di ers if the queue is empty is called special service, and the concept is suitable for the analysis of a single priority level in isolation. The priority queue is converted to as many separate G G 1-Special Service queues as many
