M/g/ polling systems with random visit times
We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this paper is the first in which an M/G/-type polling system is analysed. For this polling model, we derive the probability generating function and expected value of the queue lengths, and the Laplace-Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximises the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is independent of the number of customers present at the various queues at the start of the cycle.
Georgia Institute of Technology,
H. Milton Stewart School of Industrial & Systems Engineering,
765 Ferst Drive, Atlanta GA 30332-0205, USA.
Department of Statistics and Operations Research,
School of Mathematical Sciences,
Raymond and Beverly Sackler Faculty of Exact Sciences,
Tel Aviv University, Tel Aviv 69978, Israel.
Keywords: M/G/ queue, polling systems, optimal visit times, dynamic polling schedules, Hamiltonian tours, index rules, road-traffic control
AMS 2000 Subject Classification: Primary 60K30. Secondary 90B20, 68M20, 68K25, 60E10
A typical polling system consists of a number of queues, attended by a single server in a cyclic fashion. There is a huge body of literature on polling systems that has developed since the late 1950s, when the papers of Mack et al. [12, 13] concerning a patrolling repairman model for the British cotton industry were published. Rather than giving a partial overview of the literature, we refer the interested reader to the following books, surveys, and papers on polling systems: Takagi [17, 18, 19], Boxma and Groenendijk , Levy and Sidi , Yechiali , Borst , Eliazar and Yechiali , Nakdimon and Yechiali .
Polling systems have been used as a central model for the analysis of a wide variety of applications in the areas of repair problems [12, 13], telecommunication systems , road traffic control , computer networks , multiple access protocols , multiplexing schemes in ISDN , satellite systems , flexible manufacturing systems , and the like.
In many of these applications, as well as in most polling models, it is customary to control the amount of service given to each queue during the server’s visit. Common service policies are the exhaustive, gated, globally gated and limited regimes. Under the exhaustive regime, at each visit the server attends the queue until it becomes completely empty, and only then is the server allowed to move on. Under the gated regime, the only customers served during a visit are the ones who are present when the server enters (polls) the queue, while customers arriving when the queue is attended will be served during the next visit. The globally gated regime, introduced by Boxma, Levy and Yechiali , is a modification of the gated one: the only customers served during a visit are those who are present at the beginning of a cycle. Finally, under the -Limited service discipline only a limited number of jobs (at most ) are served at each server’s visit to each queue. These service policies imply that the duration of the visit time in a polled queue is a function of the number of customers present there at a given moment (such as the beginning of the cycle or the moment the server enters the queue).
In this paper, we analyse a polling system that differs in two ways from the classical polling model. Rather than considering a single server providing service to customers at the various queues, we assume that an infinite number of servers is moving as a single group between the queues. Moreover, the service policy we study is independent of the queue length. We assume that the group of servers visits each queue for a (possibly random) amount of time that is independent of everything else and which has a distribution that may vary per queue. We further assume that the arrival process of customers to each queue is Poisson and that the service time distribution for customers in each queue is general. To the best of our knowledge, this paper is the first in which an M/G/-type polling system is analysed.
The specific application that raised our attention and led us to this model is in the field of road traffic control. Polling models for road traffic are typically along the lines of the classical polling system; namely, they involve a single server rotating around a number of queues. Other assumptions that are typically being made for such models include deterministic service times (i.e. the amount of time that a car needs to pass a traffic light after possibly standing in the queue) and deterministic visit times (i.e. the time the traffic light remains green); see, for example, van der Heijden . Although these models provide fairly good approximations of reality, such assumptions fail to capture the variation both in service times and in visit times. Cars do not need the same amount of time to cross a segment of the road; the ones standing ahead in the queue will inevitably need less time and those that arrive while the queue is empty and the traffic light is still green will not even require the additional time incurred by acceleration. Moreover, recent developments in the technology of traffic lights has led to the design of traffic lights that do not turn green unless a queue is formed, and turn red either when the queue is empty or after a maximum amount of time, which may also vary within a day. As a result, in this paper we provide a framework for studying road traffic control under less restrictive assumptions. We propose an infinite-server polling system, which models the behaviour of traffic: while the traffic light is green all cars present in the queue or approaching the traffic light proceed (receive service) and the time they need to complete service is assumed to be a random variable following a general distribution. Furthermore, we assume that the time the traffic light is green (visit time) is random, although our results are directly applicable in case of deterministic visit times or, more generally, in case the visit times follow a discrete distribution taking positive values.
A common approximation to road traffic is to consider the traffic as fluid passing through the road. This approximation is fairly accurate when the traffic is relatively high. Mathematically, high traffic can be modelled by assuming that the arrival rate of customers at each of the queues tends to infinity. The study of such a model provides insights at the queue length (and thus the congestion of a junction) under heavy load. In this paper though we do not study the evolution of the system under heavy load. We assume that the arrival rate at each queue is fixed. This assumption is usually made for the standard polling systems and provides a reasonable approximation to normal traffic conditions.
The rest of the present paper is organized as follows. Section 2 introduces the model, gives further notation, and describes formally the evolution of the system. In Section 3 we compute recursively the first moment and the probability generating function of the queue length distributions at a polling instant. Later on, in Section 4 we derive the mean and the Laplace-Stieltjes transform of the sojourn time of a customer arriving at queue , and we show how these expressions simplify in the special case where both the service time and the visit time at queue are exponentially distributed. Based on the results derived up to that point, in Section 5 we give some numerical results. Specifically, we examine numerically the effect of the first two moments of the visit and service times on the sojourn time of an arbitrary customer. These numerical results indicate that there is an optimal value for the mean visit time to the various queues that minimises the mean sojourn time of an arbitrary customer. In Section 6 we investigate how we can optimise the visit order of the servers at the various queues so that the expected throughput of the system is maximised. It emerges that even when considering semi-dynamic control policies, in which the group of servers plans a new route for each cycle, the optimal visiting order that maximises the expected throughput per cycle is fixed for all cycles. In other words, because of the infinite number of servers, information regarding the queue lengths of all queues at the beginning of a cycle has no effect on the choice of the optimal strategy.
2 Model description and notation
We consider a polling system with infinite-buffer queues attended by a group of ample number of servers that visits the queues in a fixed cyclic fashion. We index the queues by in the order of the servers’ movement. We shall refer to the polling instant of queue as the moment when the servers enter that queue. When visiting queue , the group of servers continues working at this queue for units of time, and acts there as an M/G/ queue. We assume that the visit times are independent, identically distributed (i.i.d.) random variables.
Customers arrive at all queues according to independent homogeneous Poisson processes with rate for queue . After completing their service time, customers leave the system. The service time of each individual customer at queue is denoted by . It is assumed that all service times in one queue are i.i.d. random variables, which are mutually independent of all service times at any other queue. At the end of a visit to queue , the group of servers moves to queue , incurring a switch-over time and a realisation of is drawn. We assume that is a sequence of independent random variables. The total switch-over time during a full cycle is , and the length of a full cycle is denoted by the random variable . We assume that all random variables so far are mutually independent.
During the visit time of the group of servers to queue , a customer present at queue at the polling instant of that queue will successfully complete his service with probability . We assume that if the service of a customer of queue is not completed during a single visit, then at the next visit a new service time will be drawn from the service time distribution of for that particular customer.
For a generic random variable , we denote its first two moments by and , respectively. Thus, for example, is the mean visit time of the servers at queue . By convention, , and similarly for the product operator. All further notation will be introduced when it is first used.
Law of motion
Let , , denote the number of customers in queue at the moment when queue is polled and let denote the number of Poisson arrivals to queue during a time interval of length . The law of motion describing the evolution of the system when the server moves from queue to queue connects to and is given by
where for all , is an i.i.d. copy of , Binom is a binomial random variable with parameters and , and Poisson is a Poisson random variable with rate
Note that from (2.1) we see that for all , the random variables are independent of and , which is evident, considering that the number of customers in a queue at the beginning of a visit does not depend on the length of the upcoming visit time or switch-over time.
The relation for is straightforward. The number of customers at queue at polling instant of queue equals the number of customers that were there at polling instant of queue plus all customers that arrived during the visit time of queue and the switch-over time from queue to queue .
For , the relation is more involved. When the servers start polling queue they encounter customers. After time units, only a binomial number of customers out of the initial is still present. The probability that a single customer does not complete his service after time units is . In addition, there is a stream of new arrivals to queue . The number of customers present at time in an M/G/ queue (starting with zero customers at time ) is Poisson distributed with rate , as it is given above; see Takács . The last term at the right-hand side of (2.1) incorporates the customers that arrived at the queue during the switch-over time from queue to queue .
We shall employ this relation to derive the mean queue length and the probability generating function of all queues at a polling instant.
3 Queue lengths at polling instants
One of the main tools used in the analysis of polling systems is the derivation of a set of multi-dimensional probability generating functions of the number of jobs present in the various queues at a polling instant of queue . The common method is to derive the probability generating function of a given queue at some polling instant in terms of the probability generating function of the same queue at the previous polling instant. Then, from the set of (implicit) dependent equations of the unknown probability generating functions one can obtain expressions which allow for numerical calculation of the mean queue length at each queue. These equations simplify significantly for several cases of the distribution of the visit times. In this section, we use the law of motion (buffer occupancy), which is given by Equation (2.1) and apply this technique to compute recursively the first moment and the probability generating function of the queue length distributions at a polling instant.
3.1 Mean queue length
From (2.1) we have the following relation for the mean queue length of queue at two consecutive polling instants.
where . Summing (3.1) over we obtain
Indeed, in steady state, the mean number of jobs in queue at a polling instant equals the fraction of jobs left behind at the end of the previous visit, plus the mean number of arrivals during the cycle time out of queue , which is , plus the mean number of customers in a M/G/ queue at time . The mean queue length of queue at polling instant of queue is easily derived from (3.1), yielding
For example, suppose that is exponentially distributed with parameter . Then,
Thus, . So, in particular, if is also exponentially distributed with parameter , then we have that , and the mean queue length of each queue can now easily be computed recursively from (3.3).
3.2 Recursive relation for the generating function
Define the generating function of the queue length of all queues at polling instants of queue as . Then, from (2.1) we have that
By conditioning on the vector , on , and on , Equation (3.4) becomes
Since the number of arrivals at any queue during a fixed amount of time is independent of the number of arrivals at any other queue during the same given period, we have that
Therefore, we have
Likewise, we obtain that
or in other words,
For the last term of the right-hand side of (3.5) we have that
which yields that
Recall that for all , the random variables are independent of and . Therefore, Equation (3.10) becomes
where denotes the Laplace-Stieltjes transform of the random variable . Evidently, if follows a discrete distribution, the above expression simplifies significantly. Note that the mean queue length at a polling instant (3.3) can also be obtained by differentiating Equation (3.12).
Applying similar techniques, we can also derive the probability generating function of the number of customers at the end of a visit at queue . If we denote by the number of customers in queue at the moment when the service at queue is completed, then the law of motion describing the evolution of the system is given by
Also note that the expected value of can be easily computed from (3.3) by observing that for all , , while for the -th queue we have that .
4 Sojourn time
Let the sojourn time of a customer at queue be denoted by . We compute its expected value (and thus, by Little’s law, also the mean queue length of queue at an arbitrary moment), and we derive the Laplace-Stieltjes transform of . As stated before, for each queue we assume that if the service of a customer is not completed during a visit, then, for the next visit at that queue, a new service time will be resampled for the same customer from the service time distribution of that queue.
4.1 Mean sojourn time
Recall that the cycle time is given by . In order to derive the mean sojourn time of a customer arriving at queue , we shall need some further notation. Denote by the residual visit time of the group of servers at queue and by the cycle time except the time spent serving queue , i.e. . Similarly, represents the residual cycle time excluding the visit time of queue . That is, measures the length of time from a random moment after leaving queue until the next polling instant of queue . Furthermore, let be a family of i.i.d. random variables distributed like , and be a (shifted) geometric random variable with success probability , i.e. , for all integer . One should observe here that is a stopping time as it is the first time when the service time of a customer in queue is less than or equal to the visit time at that queue; that is, , where and are i.i.d. copies of and respectively. Similarly, a second index is added to a random variable, every time that we explicitly need to indicate that an independent copy is considered. Then the sojourn time of a customer at queue is given by
Note that the probability of an arrival occurring during the visit time of queue is , i.e. the expected visit time of queue over the expected cycle time, and similarly for the other two events. Therefore, from (4.1) we obtain that the expected sojourn time of a customer of queue is given by
In order to compute the second conditional expectation appearing at the right-hand side of the above equation, we think as follows. For cycles, the service of the customer is not completed during that visit because for every visit , while at the st visit the service is completed within that cycle. Therefore, define
and observe that
where in the second equality we used Wald’s equation.
For the third conditional expectation appearing at the right-hand side of (4.2), we have that
Summarising the above, we have that
In Section 5 we shall illustrate through an example the effect of the first two moments of the visit time and the service time on the mean sojourn time of an arbitrary customer.
4.2 The Laplace-Stieltjes transform
We now derive the Laplace-Stieltjes transform of the sojourn time of a customer of queue . We first rewrite Equation (4.1) in terms of the Laplace-Stieltjes transforms of all variables involved (cf. (4.5)), and thus we get that
We rewrite several of the terms appearing above as follows. The distribution function of is given by
Similarly, we have that
which implies that
where denotes the Laplace-Stieltjes transform of the random variable . Moreover,
where . Likewise, we have that
Clearly, from the expression above, one can retrieve Equation (4.5) for the mean sojourn time of a customer of queue .
The transforms appearing in (4.12) may be cumbersome to compute when the service times or the visit times are generally distributed. However, when both and follow a phase-type distribution, all transforms can be computed explicitly since the class of phase-type distributions is closed under finite minima. Phase-type distributions are widely used in computations. The class of phase-type distributions is dense (in the sense of weak convergence) in the class of all distributions on (cf. [2, Propositions 1 and 2]). As an example, we will derive the Laplace-Stieltjes transform of the sojourn time of a customer of queue , as well as its mean, in case both the visit time and the service time at queue are exponentially distributed.
4.3 A special case
Let the service time and the visit time at queue be exponentially distributed with rates and respectively. Then all terms appearing in (4.5) can be easily computed in terms of and . For example,
and . Thus, (4.5) becomes
Similarly, (4.12) reduces to
Since we have that the previous expression reduces to
Similar expressions can be easily derived in case both the visit times and the service times follow some phase-type distribution, such as Gamma, hyperexponential, or Coxian distributions.
5 Numerical results
This section is devoted to some numerical results. In particular, we want to examine numerically the effect of the first two moments of the visit and service times on the sojourn time of an arbitrary customer. In all examples, we make the following assumptions. We consider a polling system with two queues. The arrival rate at the first queue is and at the second queue it is . The service time and the visit time at the first queue are exponentially distributed with rates and respectively. Moreover, the total mean switch-over time is taken to be , while its second moment is assumed to be zero. In all figures that follow, we plot the mean sojourn time of an arbitrary customer, which is estimated by .
In Figures 5 and 5 we investigate the effect of the first two moments of the service time at the second queue on the mean sojourn time of an arbitrary customer. For these plots, the visit time at the second queue is considered to be exponentially distributed with rate . For various values of the squared coefficient of variation of the service time at the second queue, which is denoted by , we plot in Figure 5 the mean sojourn time of an arbitrary customer versus the mean service time . The squared coefficient of variation of the service time is chosen to be comparable to the squared coefficient of variation of the (exponentially distributed) visit time, which is equal to 1. In Figure 5, we plot the mean sojourn time of an arbitrary customer versus for three values of , which again are chosen to be comparable to .
For each case of , we fit a mixed Erlang or hyperexponential distribution to and , depending on whether the squared coefficient of variation is less or greater than one; see, e.g., Tijms . So, if for some , then the mean and squared coefficient of variation of the mixed-Erlang distribution