1 Introduction
Consider a single server system with multiclass customers. Some commonly used scheduling policies in such multiclass system are DPS, DROS, GPS and WFQ. A quick overview of these policies is as follows. Policies like GPS and DPS are variants of the processor sharing policy where the server can serve multiple customers from the system simultaneously. In case of GPS, a separate queue is maintained for each customer class and the total service capacity of the server is shared among customers of the different classes in proportion to predefined weights The GPS scheduling policy is often considered as a generalization of the headofline processor sharing policy (HOLPS) as described in [Aalto07, Demers90, KMK]. (Refer [Fayolle79, Morrison93] for details about HOLPS). As a generalization of HOLPS, GPS maintains a FIFO scheduling policy within the queue of each class and only the headofline customers of different classes are allowed to share the processor. The share of the server for a headofline Class customer is proportional to the weight and is independent of the number of other customers in the queue. The service rate received by the customer is precisely given by where if the queue has at least one class customer and otherwise. Refer Parekh and Gallager [Parekh93], Zhang et al. [Zhang94] for an early analysis of the model.
In case of DPS, the total service capacity is shared among all the customers present in the system and not just among the headofline customers of different classes. The share of the server for a customer of a class is not only in proportion to the class weight, but also depends on the number of multiclass customers present in the queue. In particular, a Class customer in the system is served at a rate of where denotes the number of Class customers in the system. The DPS system was first introduced by Kleinrock [Kleinrock672] and subsequently analyzed by several authors [Aetal05, Fayolle80, Haviv072, Haviv97, Kim04]. See [AAA06] for a survey of various results on DPS.
The DROS and WFQ scheduling policies are also characterized by an associated weight for each customer class. However these policies are not a variant of the processor sharing policies and hence their respective server can only serve one customer at a time. The DROS and WFQ policies differ in their exact rule for choosing the next customer. In the DROS policy, the probability of choosing a customer for service depends on the weights and the number of customers of the different classes in the queue. A Class
customer is thus chosen with a probability of where denotes the number of Class customers waiting in the system for service. DROS policy is also know as relative priority policy and was first introduced by Haviv and wan der Wal [Haviv97]. For more analysis of this policy we refer to [Haviv07, Kim11]. In the WFQ policy, a separate queue for each class is maintained and the next customer is chosen randomly from among the headofline customers of different classes. As in case of the GPS scheduling policy, a FIFO scheduling policy is used within each queue for a class. WFQ can be seen as a packetised version of GPS and the probability of choosing a headofline Class customer for service is given by where is as defined earlier. Refer Demers [Demers90] for the detailed analysis of the WFQ policy.It is interesting to note that for a Class customer, the service rate received in DPS and the probability of being chosen next for service in case of DROS is given by Similarly, the service rate received in GPS and the probability of being chosen next for service in case of WFQ is This similarity in the scheduling rules motivates us to compare the tail of the waiting time and sojourn time distributions of the multiclass customers with these scheduling policies. Assuming identically distributed service requirements for all customers, we will show that the tail of the waiting time distribution of a Class customer in a system with DROS (WFQ) scheduling policy is times the tail of the sojourn time distribution of any Class customer with DPS (resp. GPS) scheduling policy. This is a generalization of [Borst03], where the equivalence has been established between singleclass processor sharing and random order service discipline.
Organization: In the next section, we introduce a generalized notion of multiclass processor sharing (MPS) and random order service (MROS) policies. The DPS, GPS, DROS and WFQ policies will turn out to be special cases of MPS and MROS. In Section 3, we show that the tail of the sojourn time distribution of a Class customer with MPS scheduling is equivalent to the tail of the waiting time distribution of a Class customer with MROS policy. As a special case, this proves the mentioned equivalences among the four multiclass scheduling policies.
Notation: We use to denote the total number of customer classes in the system. Let denote the arrival rate for a Class customer, Let and Further, let denote a weight parameter associated with a Class customer.
Assumptions:
We assume that the service requirement of each customer is independent and exponentially distributed with rate
. Thus the service requirements are independent of their class. We also assume that customers from different classes arrive according to independent Poisson processes. For the purpose of stability, we assume that2 Generalized multiclass scheduling policies
In this section, we will describe two multiclass scheduling policies that are a generalization of policies such as DPS, DROS, GPS and WFQ. The two policies are based on the processor sharing and random order service mechanism and will be labeled as MPS and MROS respectively.
The MPS scheduling policy is a multiclass processor sharing policy where the server can serve multiple customers simultaneously. A separate queue for each customer class is maintained and a FIFO scheduling policy is used within each queue of a class. The MPS scheduling policy is parameterized by a vector
that characterizes the maximum permisible number of customers of each class that can be served simultaneously with other customers. We shall henceforth use the notation MPS when we talk about an MPS policy with parameter Let denote the instantaneous number of Class customers in the queue and denotes the corresponding state in the MPS system. Let denote the number of Class customers under service when the state of the MPS system is . Then, clearly In other words, if then all the Class customers present in the queue are being served simultaneously for . However if then only the first customers of Class in its queue are served simultaneously. It should be noted that due to the FIFO policy within each queue of a class, only the first customers in the queue are served at any time. To lighten some of the notation, we shall drop the dependence on and use only when the context is clear. For an MPS scheduling policy in state the service rate received by a particular Class customer which is in service is given by When for the corresponding scheduling policy will be denoted by MPS In this case, and therefore MPS corresponds to the DPS scheduling policy. Similarly if then MPS corresponds to the GPS scheduling policy where only the headofline customers of each class can be served.In a similar manner, we can define the MROS scheduling policy where denotes the vector of multiclass customers from which the subsequent customer is chosen for service. As in case of the MPS policy, note that a separate FIFO queue for each customer class is also maintained for the MROS system. At any given time, the first customers are candidates for being chosen for service while the remaining customers have to wait for their turn. Note that the state for an MROS scheduling policy denotes the vector of waiting multiclass customers present in the system. In the MROS system, a Class customer within the first customers in its queue will be chosen next for service with probability As in case of the MPS scheduling, MROS corresponds to the DROS policy whereas MROS corresponds to the WFQ policy.
Remark 1
A policy closely related to the MPS discipline is the limited processor sharing (LPS) policy. LPS is a single class processor sharing policy parametrized by an integer where denotes the maximum number of customers that can be served simultaneously. Here corresponds to the processor sharing policy while corresponds to FCFS policy. LPS can also be viewed as a special case of the MPS policy when there is a single service class for the arriving customers. See [AviItzhak88, Zhang11] more more details about the LPSc policy.
Having introduced the generalized multiclass scheduling policies, we shall now establish an equivalence relation between the tail of the sojourn time distribution of a Class customer in MPS system with the tail of the waiting time distribution of a Class customer in MROS system.
3 Comparing the sojourn and waiting time distributions in MPS and MROS
The analysis in this section is inspired from that in [Borst03] where a similar result is established for the case of a single class of population. For a given state of define
Let random variable
denote the conditional sojourn time experienced by an arriving Class customer that sees the system in state . The corresponding unconditional random variable will be denoted by We shall occasionally use the notation to denote the MPS system with customers. Along similar lines, let the random variable denote waiting time (time until chosen for service) experienced by an arriving Class customer that sees the MROS system in state i.e., it sees a vector of waiting customers in the system. This system will be often denoted as and the unconditional random variable will be denoted by Let anddenote the probability distribution of the random variables
and respectively. (The dependence of these distributions on have been suppressed for notational convenience.) We now state the main result of this paper.Theorem 1
for
As in [Borst03], our aim is to first provide a coupling with the corresponding law denoted by such that

and

The second requirement will help us to show that the two distributions and are equal. This follows from the coupling inequality (see [Lindvall02] for more on coupling inequalities)
(1) 
Such a coupling is precisely obtained as follows.
Consider two tagged Class customers and that arrive to a and a system respectively. This means that at the arrival instant of customer X in the system, there are Class customers already present in the system. Similarly, at the arrival instant of customer Y in there are customers of Class that are waiting for service in the queue. Recall that where in the MPS system denotes the number of Class customers that are receiving service. In the MROS system, denotes those (waiting) Class customers from which the next customer could be chosen. Note that since with the arrival of customer the system has customers. Similarly, with the arrival of customer the system has customers of which one customer is in service and the remaining customers (including customer Y) are waiting for service. We will now specify the rule for forming the required coupling. Since the customers can be distinguished by their class index and also the position in their respective queues, we couple the customers in with the waiting customers in the system based on their class and queue position. The coupling must be such that the coupled customers belong to the same class and invariably have the same queue position in their respective queues. It goes without saying that the tagged customers and are also coupled. As in [Borst03], we also couple the subsequent arriving customers and let denote i.i.d random variables with an exponential distribution of rate These random variables correspond to service times of a customer in service in
At the service completion epoch, pick a pair of coupled customers randomly. This random picking is with a distribution such that the chosen pair is of Class
with probability When the randomly chosen pair is of Class , a class customer departs from the MPS system while such a customer is taken for service in the MROS system. This process is repeated till the tagged pair leaves the system. Clearly, this joint probability space is so constructed that the random variables –a.s. From Eq. (1), this implies that(2) 
Now let random vectors (resp. ) denote the vector of multiclass customers present in the system in steady state (resp. waiting in the system in steady state in MROS). The subscript in is used to indicate a busy server. Since the arrival process is Poisson, the unconditional probabilities are given by the following
Similarly, we have
(4) 
Now if is true, then from Eq. (2), the statement of the theorem follows and this would complete the proof. In the following lemma, we shall prove that indeed
Lemma 1
for such that
We first simplify the notations as follows. Let and Let denote the probability that the MROS system has no customers and is idle. The statement of the lemma now requires us to prove that To prove this result, consider the balance equation for the MPS system where shall denote the stationary invariant distribution for the system. The assumption implies that the underlying Markov process is ergodic and hence the stationary distribution is unique. For such that the global balance equations for the MPS system are
Now since ∑_i=1^N(βi(¯n)pi∑j=1Npjβj(¯n) ) = 1, the balance equations can be written as
Similarly, the global balance equations for the MROS system are as follows for such that
(6)  
Additionally, the idle system should satisfy
(7) 
where is the probability that the system is empty. Now again, the assumption implies that the underlying Markov process is ergodic and hence the stationary distribution is also unique. Therefore to prove the lemma, it is sufficient to check if the global balance equations for the MROS system given by Eq. (6) are satisfied when
Now from Eq. (6) and assuming that we have
The last equality follows from Eq. (3) after dividing throughout by Similarly,
(8)  
Here the third equality is from the fact that is the probability that the MPS system is empty. Clearly, substituting satisfies the balance equations for the MROS system. Since is the unique invariant distribution, the statement of the lemma follows.
We now have the following corollary that establishes the desired equivalence between DPS (GPS) and DROS (resp. WFQ) scheduling policies. Note that the result is true only for the case when all customers have identically distributed service requirements. The equivalence result is not true in general when the customer classes differ in their service requirements.
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