Mixed gated/exhaustive service in a polling model with priorities

Queueing Systems, Dec 2009

In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.

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Mixed gated/exhaustive service in a polling model with priorities

M.A.A. Boon 0 I.J.B.F. Adan 0 0 The research was done in the framework of the BSIK/BRICKS project, and of the European Network of Excellence Euro-FGI. M.A.A. Boon ( ) In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers. 1 Introduction There are three ways in which one can introduce prioritisation into a polling model. The first type of priority is by changing the server routing such that certain queues are visited more frequently than other queues [6, 19]. This type of prioritisation is quite common in wireless network protocols. A second type of prioritisation is through differentiation of the number of customers that are served during each visit to a queue. This type of prioritisation is inflicted through the usage of different service disciplines. For example, one can serve all customers in a queue before switching to the next queue (exhaustive service), or one can limit the amount of customers that are served to, e.g., only those customers present at the arrival of the server at the queue (gated service). Typically, this will have a negative impact on the waiting times of the customers in queues that are not served exhaustively. The third way of introducing priorities is by changing the order in which customers are served within a queue, which is a popular technique to improve performance of production systems, cf. [2, 23]. The present paper introduces a new service discipline, referred to as mixed gated/exhaustive service, that combines the last two types of prioritisation. In the polling model considered in the present paper a single server visits N queues in a fixed, cyclic order. Some, or even all, of the queues contain two types of customers: high and low priority customers. For these queues we introduce a new service discipline, called mixed gated/exhaustive service based on the priority level of the customer. A polling system with high and low priority customers in a queue with purely gated or exhaustive service has been studied in [1, 2]. The mixed gated/exhaustive service discipline can be considered as a mixture of these two service disciplines where low priority customers receive gated service and high priority customers receive exhaustive service. A more detailed description is given in Sect. 2. Since the number of customers served during one visit in a queue with gated service is different from the number served during a visit with exhaustive service, the mixed gated/exhaustive service discipline introduced in the present paper combines the second and the third type of prioritisation. A variation of the model under consideration, namely a polling system where low priority customers are served only if there are no high priority customers present in any of the queues, has been studied in [12]. Polling models have been studied for many years and because of their practical relevance many papers on polling systems have been written in a mixture of application areas. The survey of Takagi [20] on polling systems and their applications from 1988 is still very valuable, although the last couple of years interest in polling models has revived, partly triggered by many new applications. The motivation for the present paper is to present a service discipline that combines the benefits of the gated and exhaustive service disciplines for priority polling models. The specific application that attracted our attention is in the field of logistics. Consider a make-to-order production system with a single production capacity for multiple products. In many firms encountering this situation, the products are produced according to a fixed production sequence. The production capacity, where the production orders queue up, can be represented as a polling model by identifying each product with a queue and the demand process of a product with the arrival process at the corresponding queue. For a more detailed description of fixed-sequence strategies in the context of maketo-stock production situations, see [24]. In the context of this production setting, the situation with two or more priority levelsas studied in detail in the present paper is oftentimes encountered in practice, where production departments have to supply both internal and external customers, the latter of which is commonly given a preferential treatment. A different application stems from production scheduling in flexible manufacturing systems where part types are often grouped with other types sharing (almost) similar characteristics, such that no change of machine configuration, i.e. setup time, is required when switching between these part types (see, e.g., [17]). Since no setup time is required to switch between these types, it can be seen as customers of different types being served in the same queue. The introduction of priorities can be useful to efficiently differentiate between different parts grouped within one queue. These two applications make the practical relevance of the inclusion of multiple priority levels in the studied polling model evident. Finally, we should keep in the back of our mind that the results of the present paper are certainly not limited to these production settings, but may be used in many other fields where polling models arise, such as communication, transportation and health care (e.g., surgery procedures where an urgency parameter is assigned to each patient). The present paper is structured as follows: first we discuss the model in more detail and we determine the generating functions (GFs) of the joint queue length distribution of all customers at visit beginnings and completions of each queue. In Sect. 4 we determine the LaplaceStieltjes Transforms (LSTs) of the distributions of the cycle time, visit times and intervisit times. These distributions are used to determine the marginal queue length distributions and waiting time distributions of high and low priority customers in all queues. The LST of the waiting time distribution is used to compute the mean waiting time of each customer type. A pseudo-conservation law for these mean waiting times is presented in Sect. 7. Furthermore, we introduce some numerical examples to illustrate typical features of a polling model with mixed gated/exhaustive service. Finally, we discuss possible extensions and future research on the topic. 2 Notation and model description The model considered in the present paper is a polling model which consists of N queues, labelled Q1, . . . , QN . Throughout the whole paper all indices are modulo N , so QN+1 stands for Q1. The queues are visited by one server in a fixed, cyclic order: 1, 2, . . . , N , 1, 2, . . . . The switch-over time of the server from Qi to Qi+1 is denoted by Si with LST i (). We assume that all switch-over times are independent and at least one switch-over time is strictly greater than zero. Each queue contains two customer types: high and low priority customers, although the analysis allows any number (greater than zero) of customer types per queue. High priority customers in Qi are called type iH customers and low priority customers in Qi are called iL customers, i = 1, . . . , N . Type iH customers arrive at Qi according to a Poisson process with intensity iH , and type iL customers arrive at Qi according to a Poisson process with intensity iL. The service times of type iH and iL customers are denoted by BiH and BiL, with LSTs iH () and iL(). All service times are assumed to be independent. We introduce the notation iH = iH E(BiH ) and similarly iL = iLE(BiL). The total occupation rate of the system is = iN=1 i , where i = iH + iL is the fraction of time that the server visits Qi . Service of the customers is gated for low priority customers and exhaustive for high priority customers. In more detail: each queue actually contains two lines of waiting customers: one for the low priority customers and one for the high priority customers. At the beginning of a visit to Qi , a gate is set behind the low priority customers to mark them eligible for service. High priority customers are always served exhaustively until no high priority customer is present. When no high priority customers are present in the queue, the low priority customers standing in front of the gate are served in order of arrival, but whenever a high priority customer enters the queue, he is served before any waiting low priority customers. Service is non-preemptive though, implying that service of a type iL customer is not interrupted by an arriving type iH customer. The visit to Qi ends when all type iL customers present at the beginning of this visit are served and no high priority customers are present in the queue. Notice that if the arrival intensity iH equals 0, then Qi is served completely according to the gated service discipline. Similarly we can set iL = 0 to obtain a purely exhaustively served queue. Both the gated and the exhaustive service disciplines fall into the category of branching-type service disciplines. These are service disciplines that satisfy the following property, introduced by Resing [16] and Fuhrmann [10]. Property 2.1 If the server arrives at Qi to find ki customers there, then during the course of the servers visit, each of these ki customers will effectively be replaced in an i.i.d. manner by a random population having probability generating function hi (z1, . . . , zN ), which can be any N -dimensional probability generating function. If Qi receives gated service, we have hi (z1, . . . , zN ) = i ( jN=1 j (1 zj )), where i () denotes the service time LST of an arbitrary customer in Qi , and i denotes his arrival rate. For exhaustive service hi (z1, . . . , zN ) = i ( j =i j (1 zj )), where i () is the LST of a busy period distribution in an M/G/1 system with only type i customers, so it is the root in (0, 1] of the equation i () = i ( + i (1 i ())), 0 (cf. [7, p. 250]). Property 2.1 is not satisfied if Qi receives mixed gated/exhaustive service, because the random population that replaces each of these customers depends on the priority level. In the next section we circumvent this problem by splitting each queue into two virtual queues, each of which has a branching-type service discipline. This equivalent polling system satisfies Property 2.1, so we can still use the methodology described in [16] to find, e.g., the joint queue length distribution at visit beginnings and completions. All other probability distributions that are derived in the present paper can be expressed in terms of (one of) these joint queue length distributions. 3 Joint queue length distribution at polling epochs In the present section we analyse a polling system with all queues having two priority levels and receiving mixed gated/exhaustive service, but in fact each queue would be allowed to have any branching-type service discipline. Denote the GF of the joint queue length distribution of type 1H , 1L, . . . , N H , N L customers at the beginning and the completion of a visit to Qi by respectively Vbi (z1H , z1L, . . . , zN H , zN L) and Vci (z1H , z1L, . . . , zN H , zN L). As discussed in the previous section, the polling model under consideration does not satisfy Property 2.1, which often means that an exact analysis is difficult or even impossible. For this reason we introduce a different polling system that does satisfy Property 2.1 and has the same joint queue length distribution at visit beginnings and endings. The equivalent system contains 2N queues, denoted by Q1H , Q1L , . . . , QN H , QN L . The switch-over times Si , i = 1, . . . , N , are incurred when the server switches from QiL to Q(i+1)H ; there are no switchover times between QiH and QiL . Customers in this system are so-called smart customers, introduced in [4], meaning that the arrival rate of each customer type depends on the location of the server. Type iH customers arrive in QiH according to arrival rate iH unless the server is serving QiL . When the server is serving QiL , the arrival rate of type iH customers is 0. The reason for this is that we incorporate the service times of all type iH customers that would have arrived during the service of a type iL customer, in the original polling model, into the service time of a type iL customer. In our alternative system, type iL customers arrive with intensity iL and have service requirement BiL with LST iL(). There is a simple relation between BiL and BiL, expressed in terms of the LST: BiL is often called completion time in the literature, cf. [21], with mean E(BiL) = E(BiL) . Service is exhaustive for Q1H , Q2H , . . . , QN H and synchronised gated for 1iH Q1L , Q2L , . . . , QN L , the gate of QiL being set at the visit beginning of QiH . The synchronised gated service discipline is introduced in [15] and does not strictly satisfy Property 2.1. However, it does satisfy a slightly modified version of Property 2.1 that still allows for straightforward analysis; see [3] for more details. During a visit to QiL only those type iL customers are served that were present at the previous visit beginning to QiH . The joint queue length distribution at a visit beginning of QiH in this system is the same as the joint queue length distribution at a visit beginning of Qi in the original polling system. Similarly, the joint queue length distribution at a visit completion of QiL is the same as the joint queue length distribution at a visit completion of Qi in the original polling system. In terms of the GFs: Vbi (z) = VbiH (z), Vci (z) = VciL (z), where z is a shorthand notation for the vector (z1H , z1L, . . . , zN H , zN L). The GFs of the joint queue length distributions at a visit beginning and completion of QiH are related in the following manner: VciH (z) = VbiH z1H , z1L, . . . , hiH (z), ziL, . . . , zNH , zNL , j =i (jH (1 zjH ) + jL(1 zjL))). Similarly: VciL (z) = VbiH z1H , z1L, . . . , hiH(z), hiL(z), . . . , zNH , zNL , where hiL(z) = iL(iL(1 ziL) + j =i (jH (1 zjH ) + jL(1 zjL))). Note that VciH () = VbiL () since there is no switch-over time between QiH and QiL . There is a switch-over time between QiL and Q(i+1)H though: j=1 Now that we can relate Vb(i+1)H () to VbiH (), we can repeat these steps N times to obtain a recursive expression for VbiH (). This recursive expression is sufficient to compute all moments of the joint queue length distribution at a visit beginning to QiH by differentiation, but the expression can also be written as an infinite product which converges if and only if < 1. We refer to [16] for more details. 4 Cycle time, visit time and intervisit time We define the cycle time Ci as the time between two successive visit beginnings to Qi , i = 1, . . . , N . The LST of the distribution of Ci , denoted by i (), can be expressed in terms of Vbi () because the type iL customers that are present at the beginning of a visit to Qi are those type iL customers that have arrived during the previous cycle. It is convenient to introduce the notation Vbi (ziH , ziL) = Vbi (1, . . . , 1, ziH, ziL, 1, . . . , 1), where ziH and ziL are the arguments that correspond respectively to type iH and iL customers. Using this notation we can write: Vbi (1, z) = i (iL(1 z)). Hence, the LST of the cycle time distribution is: Note that E(Ci ) = E(S1)+1+E(SN ) , which does not depend on i. Higher moments of the cycle time distribution do depend on the cycle starting point. We define the intervisit time Ii as the time between a visit completion of Qi and the next visit beginning of Qi . The type iH customers present at the beginning of a visit to Qi are exactly those type iH customers that arrived during the previous intervisit time Ii . Hence, Vbi (z, 1) = Ii (iH(1 z)), where I() is the LST of the distribution of Ii . This leads to the following expression for the LST of the intervisit time distribution of Qi : The LSTs of the distributions of the cycle time and intervisit time are needed later in this paper. For the visit time of Qi , Vi , we mention the LST here for completeness but it will not be used later: 5 Waiting times and marginal queue lengths 5.1 High priority customers Ii with probability 11iHi . This leads to the following expression for the LST of the waiting time distribution of a type iH customer: The GF of the marginal queue length distribution of type iH customers can be found by applying the distributional form of Littles Law [13] to the sojourn time distribution: This leads to the following expression: (1 iH )(1 z)iH (iH (1 z)) iH (iH (1 z)) z 5.2 Low priority customers In this subsection we determine the GF of the marginal queue length distribution of type iL customers, and the LST of the waiting time distribution of type iL customers. In order to obtain these functions, we regard the alternative system with 2N queues as defined in Sect. 3. The waiting time (excluding the service time) of type iL customers in the original polling system has the same distribution as the waiting time of type iL customers in the alternative system (again excluding the service time, which is different). From the viewpoint of a type iL customer, the system is an ordinary polling system with synchronised gated service in QiL . We apply the FuhrmannCooper decomposition to the alternative polling model with 2N queues and type iL customers having completion time BiL. Using arguments similar as in the derivation of (3.7) in [3], we find the general form of the GF of the marginal queue length distribution: E zNiL (1 iL)(1 z)iL(iL(1 z)) iL(iL(1 z)) z VciL (1, . . . , 1, z, 1, . . . , 1) VbiL (1, . . . , 1, z, 1, . . . , 1) , (1 z)(E(NiL|Iend ) E(NiL|Ibegin )) where iL = 1iLiH and iL() is given by (3.1). Furthermore, NiL|Iend and NiL|Ibegin are the number of type iL customers at respectively the visit beginning and visit completion of QiL . The visit beginning corresponds to the end of the intervisit period IiL, and the visit completion corresponds to the beginning of the intervisit period. Substitution into (5.3) leads to the following expression: where we use that E(NiL|Iend ) E(NiL|Ibegin ) = iL(1 iL)E(C) = iL(1 iL )E(C), because this is the mean number of type iL customers that arrive during 1iH the intervisit time of QiL . Applying the distributional form of Littles Law to (5.4), we obtain the LST of the sojourn time distribution of type iL customers. Since the sojourn time is WiL + BiL, d and WiL = WiL, the LST of the waiting time distribution of a type iL customer immediately follows: = iL(1 iL( + iH (1 iH ()))) Vbi (iH (), iL( + iH (1 iH ()))) Vbi (iH (), 1 iL ) . The GF of the marginal queue length distribution of type iL customers can be obtained by applying the distributional form of Littles Law to the sojourn time of a type iL customer: E zNiL iL )(1 z)iL(iL(1 z) + iH (1 iH (iL(1 z)))) (1 1iH Vbi (iH (iL(1 z)), iL(iL(1 z) + iH (1 iH (iL(1 z))))) Vbi (iH (iL(1 z)), z) , 6 Moments Differentiation of the waiting time LSTs derived in the previous section leads to the following mean waiting times: E(WiH ) = E(WiL) = iH E(XiH XiL) E(Ci,res) + 1 iH iLiH E(C) E(Bi2H ) . We use a similar notation for the residual service time of a type iL customer, 2E(BiH ) the residual intervisit time, and residual cycle time. Furthermore, XiH and XiL are respectively the number of type iH and type iL customers at the beginning of a visit to Qi , so E(XiH XiL) is obtained by differentiating Vbi (ziH , ziL) with respect to ziH and ziL (and then setting ziH = ziL = 1). We now present an alternative, direct way to obtain the mean waiting time for a type iL customer by conditioning on the event that an arrival takes place in a visit period, or in an intervisit period. iL 1 + 1 iH E Ii2 + E(Vi Ii ) E(Ii,past) + E(Ii,res) In the above derivation, we use that both the past and residual intervisit times have expectation 2EE((IIi2i)) , and that if a type iL customer arrives during the visit time (with E(XiHXiL) = E NiL(Vi ) + NiL(Ii ) NiH(Ii ) = E E NiL(Vi ) + NiL(Ii ) NiH(Ii ) | Ii , Vi where Nj (T ) denotes the number of type j customers that have arrived during time T (j = iH, iL), and Vi denotes the length of a visit of the server to Qi . Hence, which coincides with the last term in (6.3). 7 Pseudo-conservation law for priority polling systems Boxma and Groenendijk [5] have shown that a so-called pseudo-conservation law holds for non-priority polling systems. We do not discuss this law in detail in the present paper, but we mention that a generalised version of this law (cf. [9, 18]) holds for systems with multiple priority levels in each queue: i=1 k=1 i=1 k=1 i=1 i=1 where S = iN=1 Si , and Ki is the number of priority levels in Qi . In this expression Zii is the amount of work at Qi when the server leaves this queue and depends on the service discipline. It is well known that for gated service, E(Zii ) = i2E(C) and for exhaustive service, E(Zii ) = 0. The pseudo-conservation law also holds for polling systems with mixed gated/exhaustive service in some or all of the queues. If Qi receives mixed gated/exhaustive service, we have Ki = 2, and E(Zii ) = iLi E(C). 8 Numerical results In order to illustrate the effect of using a mixed gated/exhaustive service discipline in a polling system with priorities, we compare it to the commonly used gated and exhaustive service disciplines. In this example we use a polling system which consists of two queues, Q1 and Q2. Customers in Q1 are divided into high priority customers, arriving with arrival rate 1H = 120 , and low priority customers, with arrival rate 1L = 140 . Customers in Q2 all have the same priority level and arrive with arrival rate 2 = 120 . All service times are exponentially distributed with mean 1. The switch-over times S1 and S2 are also exponentially distributed with mean 1, which results in a mean cycle time of E(C) = 10. The service discipline in Q2 is gated, the service discipline in Q1 is varied: gated, exhaustive and mixed gated/exhaustive. Results for a queue with two priority levels and purely gated or exhaustive service are obtained in [1]. Table 1 displays the mean and the variance of the waiting times of the three customer types under the three service disciplines. We conclude that, compared to gated service, the mixed gated/exhaustive service is a major improvement for the high priority customers in Q1, whereas the mean waiting times of the low priority customers in Q1 and the customers in Q2 hardly deteriorate. Of course in systems where 1H is quite high, the negative impact can be bigger and one has to decide exactly how far one wants to go in giving extra advantages to customers that already receive high priority. When comparing the mixed gated/exhaustive strategy to a system with purely Table 1 Numerical results for Example 1. The switch-over times S1 and S2 are exponentially distributed with mean 1. The mixed gated/exhaustive service discipline is compared to gated and exhaustive service Table 2 Numerical results for Example 1. Switch-over times are deterministic: S1 = S2 = 10 exhaustive service in Q1, we conclude that the improvement is not so much in the mean waiting time for high priority customers, but mostly in the mean and variance of the waiting time for customers in Q2. It is noteworthy that the mixed gated/exhaustive service discipline does not always have a negative effect on the mean waiting time of low priority customers in Q1, E(W1L), compared to the gated service discipline. If, for example, the switch-over times are taken to be deterministic with value 10, the mean waiting time for low priority customers is significantly less for the mixed gated/exhaustive service than for gated service, as can be seen in Table 2. Compared to gated service, type 1H customers benefit strongly from the mixed gated/exhaustive service discipline, and even type 1L customers benefit from it. The mean waiting time for customers in Q2 has increased, but only marginally. In order to get more understanding of this surprising behaviour of the waiting time of low priority customers as function of the arrival intensities 1H and 1L, we use a simplified model which leads to more insightful expressions, but displays the same characteristics as the model that was analysed in the previous paragraph. Instead of analysing a polling model, we analyse an M/G/1 queue with multiple server vacations. The queue, denoted by Q1 to use familiar notation, contains high (type 1H ) and low (type 1L) priority customers. Also here high priority customers are served before low priority customers. The service times of both customer types are exponentially distributed with mean 1. This is for notational reasons only, for this example we actually only require that both service times are identically distributed. One server vacation has a fixed length S. If the server does not find any customers waiting upon arrival from a vacation, he takes another vacation of length S, and so on. In order to stay consistent with the notation used earlier, we denote the occupation rate of high and low priority customers by respectively 1H and 1L. The total occupation rate is = 1 = 1H + 1L. Note that in this example 1H = 1H and 1L = 1L. We now compare the mean waiting times of type 1L customers in the system with purely gated service and the system with mixed gated/exhaustive service. For this simplified model, we can write down explicit expressions that have been obtained by differentiating the LSTs and solving the resulting equations. These expressions could also have been obtained by using Mean Value Analysis (MVA) for polling systems [23, 25]. S(1 + (1 21H )) Mixed G/E service: E(W1L) = (1 )(1 1H ) + 2(1 )(1 1H ) . (8.2) Now we analyse the behaviour of these waiting times as we vary 1H between 0 and , while keeping 1H + 1L = constant. Substitution of 1H = 0 shows that the mean waiting times in the gated and mixed gated/exhaustive system are equal: S(1 + 2) Mixed G/E service: E(W1L|1H ) = (1 )2 + 2(1 ) . Two interesting things can be concluded from these two equations for the case For fixed , E(W1L) in a gated system is always less than E(W1L) in a mixed gated/exhaustive system. The difference between E(W1L) in a gated system and E(W1L) in a mixed gated/exhaustive system does not depend on S. Focussing on the mean waiting time of type 1L customers only, we conclude that a gated system performs the same as a mixed gated/exhaustive system as 1L = , and that a gated system always performs better when 1L 0. For 0 < 1L < the vacation time S determines which system performs better. By taking derivatives of (8.1) and (8.2) with respect to 1H and letting 1H 0, one finds that the mean waiting time of a type 1L customer in a mixed gated/exhaustive system is less than in a purely gated system when 1H 0, if and only if S > 12+ . Since a gated system always outperforms a mixed gated/exhaustive system when 1H , for S > 12+ there must be (at least) one value of 1H for which the two systems perform the same. Further inspection of the derivatives gives the insight that in a gated system the relation between E(W1L) and 1H is a straight line, which can also be seen immediately from (8.1). In a mixed gated/exhaustive system, the relation between E(W1L) Fig. 1 Mean waiting time of type 1L customers in the polling model discussed in Example 1. For gated and mixed gated/exhaustive service, E(W1L) is plotted against 1H while keeping 1L + 1H constant. The switch-over times S1 = S2 = S/2 are deterministic and 1H is not a straight line, both the first and second derivatives with respect to 1H are strictly positive. This means that for S 12+ the gated system always performs better than the mixed gated/exhaustive system for any value of 1H > 0, and for S > 12+ the mixed gated/exhaustive system performs better than the gated system for 0 < 1H < 1H . The value of 1H can be determined analytically: From this expression we conclude that limS 1H = . Although we have studied only the vacation model, the conclusions are also valid for more general settings, like polling models with non-deterministic switch-over times, but the expressions are by far not as appealing. We visualise the findings of the present section in Fig. 1, where we show three plots of the mean waiting time of type 1L customers against 1H . The model considered is the same as in the beginning of the present section (two queues, gated service in Q2) except for the switch-over times S1 and S2, which are now deterministic. We compare gated service in Q1 to mixed gated/exhaustive service for three different switch-over times (notice that the scales of the three plots in Fig. 1 are different). In the previous example we showed that the mixed gated/exhaustive service discipline does not necessarily have a negative impact on the mean waiting times of low priority customers. In this example we aim at giving a better comparison of the performance of the gated, exhaustive and mixed gated/exhaustive service disciplines in a polling system with priorities. The polling system considered consists of two queues, each having high and low priority customers. The switch-over times S1 and S2 are exponentially distributed with mean 10. Service times of all customer types are exponentially distributed with mean 1. The arrival rates of the various customer types are: 1H = 1L = 110 , and 2H = 2L = 270 . The total occupation rate of this polling system is = 190 , and we deliberately choose a system where the occupation rates of Queue Service discipline E(WiL) E(WiH ) Var(WiL) Var(WiH ) Table 3 Expectation and variance of the waiting times of the polling model discussed in Sect. 8, Example 2 the two queues are very different, and the switch-over times are relatively high compared to the service times. The reason is that we envision production systems as the main application for the present paper (see also Sect. 1). In these applications large setup times are very common (see, e.g., [24]). Table 3 shows the mean and variance of the waiting times of all customer types of this polling system for all combinations of gated, exhaustive and mixed gated/exhaustive service. We leave it up to the reader to pick his favourite combination of service disciplines, but our preference goes out to the system with exhaustive service in Q1 and mixed gated/exhaustive service in Q2 because in our opinion the best combination of low mean waiting times and moderate variances is obtained in this system. 9 Possible extensions and variations Many extensions or variations of the model discussed in the present paper can be thought of. In this section we discuss some of them. A globally gated system The globally gated service discipline has received quite some attention in polling systems. Instead of setting the gates at the beginning of a visit to a certain queue, the globally gated service discipline states that all gates are set at the beginning of a cycle, which is the start of a visit to an arbitrarily chosen queue. The model under consideration can be analysed using similar techniques if high priority customers are served exhaustively, but low priority customers are served according to the globally gated service discipline. One would first have to build a similar model that contains 2N queues and determine the joint queue length distribution at visit beginnings and endings. The cycle time, starting at the moment that all gates are set, can be expressed in terms of the GF of the number of customers at the beginning of that cycle. Waiting times for high priority customers can be obtained using delay-cycles again, and waiting times for low priority customers can be obtained using the FuhrmannCooper decomposition. The LST of the waiting time distribution of low priority customers gets more complicated as the queue gets served later in the cycle. More than two priority levels It is possible to analyse a similar model as the one of Sect. 2, but with more than two, say Ki , priority levels in Qi . These Ki priority levels still have to be divided into two categories: high priority levels 1, . . . , ki that receive exhaustive service, and low priority levels ki + 1, . . . , Ki that receive gated service. The methodology from Sect. 5 can be used, combined with the techniques that are used to analyse a polling model with multiple priority levels, cf. [2]. A mixture of gated and exhaustive without priorities One could think of a system where each queue contains two customer classes having respectively the exhaustive and gated service discipline, but service is First-Come-First-Served (FCFS). The model is similar to the model discussed in this paper, with the exception that no overtaking takes place. Customers that are served exhaustively will not be served before any gated customers standing in front of this gate, but they are allowed to pass the gate. The joint queue length distributions at polling epochs and the cycle times are the same as for the system considered in the present paper. Since no overtaking takes place, the waiting times can be found without the use of delay-cycles. Nevertheless, analysis of the waiting times is quite tedious because a visit of a server to Qi consists of three parts. The third part is the service of exhaustive customers behind the gate, the first part is the service of the gated customers that have arrived during the previous third part and the second part is the FCFS service of both gated and exhaustive customers that have arrived during the previous intervisit time of Qi . A combination of this non-priority mixture of gated and exhaustive, and the service discipline discussed in the present paper is discussed by Fiems et al. [8]. They introduce, albeit in the different setting of a vacation queue modelled in discrete time, a service discipline where high priority customers in front of the gate are served before low priority customers waiting in front of the gate. The difference with the model discussed in the present paper, is that high priority customers entering the queue while it is being visited can pass the gate, but are not allowed to overtake low priority customers standing in front of the gate. Acknowledgements The authors wish to thank Erik Winands for his many helpful remarks and discussions. His contribution to the present paper is very much appreciated. Our gratitude also goes out to Jacques Resing who suggested the mixed gated/exhaustive service discipline. Finally, the authors thank Onno Boxma for valuable discussions and for useful comments on earlier drafts of the present paper. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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M. A. A. Boon, I. J. B. F. Adan. Mixed gated/exhaustive service in a polling model with priorities, Queueing Systems, 2009, 383, DOI: 10.1007/s11134-009-9115-z