Polling: past, present, and perspective

Abstract

This is a survey on polling systems, focussing on the basic single-server multi-queue polling system in which the server visits the queues in cyclic order. The main goals of the paper are: (i) to discuss a number of the key methodologies in analyzing polling models; (ii) to give an overview of recent polling developments; and (iii) to present a number of challenging open problems. Assumption 7The service order within each queue is First-Come First-Served (FCFS). This assumption was almost universally made in the polling literature until the work of Wierman et al. (2007). In Sect. 7, we will discuss non-FCFS service orders.Assumption 8 The times to switch from Q i , i = 1, 2, . . . , n, to the next queue are assumed to be i.i.d. random variables, generically denoted by S i , with distribution S i (·) and LST σ i (·). All switchover times are assumed to be independent of each other and of the interarrival and service times. When the switchover times between successive queues are all zero, a special situation arises. If the system has become empty after a visit to, say, Q i in the case of zero switchover times, then the server is assumed to visit queues Q i+1 , . . . , Q n (which now takes zero time) and stay in front of Q 1 (see Sect. 4). In the case of non-zero switchover times, the server is assumed to keep switching in an empty system. Assumption 9As soon as a customer has been served, it leaves the system. At some places, we briefly mention the case of customer routing; a served customer might rejoin the same queue, or join another one.

Analysis of polling models with a self-ruling server

“…Another research direction is the determination of the stability condition of the SRS disciplines. To give an indication of why it is a study on its own, consider the following counter intuitive example, where extra arrivals make a system stable: Consider a system with two queues with the following characteristics: -independent single Poisson arrival streams at the queues with rates λ 1 (specified later) and λ 2 = 2; -at Q (1) the service time is 0, p (1) = 1, that is, the server always decides that the first customer is the final one and the time the server waits idle for a customer to arrive, if any, has an exponential time length with rate ξ = 1; -the service times at Q (2) have an exponential distribution with rate μ 2 = 4, p (2) = 1, and the server leaves immediately after becoming idle, so during a visit to Q (2) , the expected number of arrivals to Q (1) is λ 1 /4 and to Q (2) 1/2.…”

“…When λ 1 = 0, Q (1) will be empty and the expected number of extra customers to Q (2) after an idle period at Q (1) is λ 2 /ξ = 2, so Q (2) is not stable since the server serves only one customer per visit to Q (2) . On the other hand, when λ 1 = 3, both queues are stable.…”

“…On the other hand, when λ 1 = 3, both queues are stable. We can see this as follows: Q (2) will always become empty when Q (1) is full since the expected number of new arrivals to Q (2) between two visit starts at this queue is λ 2 /μ 2 = 1/2; once Q (2) is empty, Q (1) will become empty. Note that, in this case, the expected number of extra customers in Q (2) after an idle period at Q (1) will be λ 2 /(λ 1 + ξ) = 1/2.…”

“…We can see this as follows: Q (2) will always become empty when Q (1) is full since the expected number of new arrivals to Q (2) between two visit starts at this queue is λ 2 /μ 2 = 1/2; once Q (2) is empty, Q (1) will become empty. Note that, in this case, the expected number of extra customers in Q (2) after an idle period at Q (1) will be λ 2 /(λ 1 + ξ) = 1/2. Since not every visit to Q (1) ends with an idle time, the expected number of new customers arriving at Q (2) in between to visit starts at Q (2) is less than one, which makes Q (2) stable.…”

“…For instance, traffic light systems, product-assembly systems, and wireless communication systems were modeled as polling systems. Good surveys on a broad class of polling models and their analysis can be found in, for example, [2,[14][15][16].…”