Performance analysis of polling systems with retrials and glue periods

Abstract

We consider gated polling systems with two special features: (i) retrials and (ii) glue or reservation periods. When a type-i customer arrives, or retries, during a glue period of the station i, it will be served in the following service period of that station. Customers arriving at station i in any other period join the orbit of that station and will retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems. When the glue periods are exponentially distributed, we obtain equations for the joint generating functions of the number of customers in each station. We also present an algorithm to obtain the moments of the number of customers in each station. When the glue periods are generally distributed, we consider the distribution of the total workload in the system, using it to derive a pseudo-conservation law which in turn is used to obtain accurate approximations of the individual mean waiting times. We also investigate the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times.

Appendix: Approximation of the mean waiting times

Below we outline a method to approximate the mean waiting times of all customer types. The arrival of a type-i customer occurs either during a glue period of station i or during any other period. At the start of the service period, the customers which will be served in the current service period are fixed. The mean length of the service period is now the same irrespective of the order in which these customers are served. Without loss of generality, we will assume that the customers who arrive during a glue period of station i are served first and then customers who retry are served.

Let $\bar{W}_i$ and $\tilde{W}_i$ denote the waiting times of type-i customers who arrive during a glue period of station i and any other period, respectively. Further, $G_{i_\mathrm{res}}$ denotes the residual time of a glue period of station i. Finally, $C_{i_\mathrm{res}}$ denotes the residual time of a non-glue period of station i. A type-i customer arriving during a glue period of station i has to wait for the residual glue period. Further, it has to wait for all the customers who arrived before it during the glue period. Therefore,

$$\begin{aligned} \mathbb {E}[\bar{W_i}]= \mathbb {E}[G_{i_\mathrm{res}}] + \rho _i \mathbb {E}[G_{i_\mathrm{res}}] = (1+\rho _i) \mathbb {E}[G_{i_\mathrm{res}}]. \end{aligned}$$

A type-i customer arriving during a non-glue period of station i has to wait for the residual non-glue period and the glue period. Then, it either gets in the queue for service or it remains in the orbit. With probability $\tilde{G}_i (\nu _i)$ it remains in the orbit and has to wait until the next visit to get served, and this repeats. Therefore, on average, it has to wait for $\tilde{G}_i (\nu _i)/(1-\tilde{G}_i (\nu _i))$ cycles before it gets into the queue for service. When it gets in the queue, it has to wait for all the type-i customers who have arrived during the glue period to be served, and then the customers who arrived before it and who will be served in the current service period (on average this number is approximately equal to the number of customers who arrived during the residual non-glue period before the arrival of the tagged customer). Therefore,

$$\begin{aligned} \mathbb {E}[\tilde{W_i}]&\approx \mathbb {E}[C_{i_\mathrm{res}}] + \mathbb {E}[G_i] + \frac{\tilde{G}_i (\nu _i)}{1-\tilde{G}_i (\nu _i)} \mathbb {E}[C] + \rho _i \mathbb {E}[G_i] + \rho _i \mathbb {E}[C_{i_\mathrm{res}}] \\&= (1 + \rho _i) \left( \mathbb {E}[C_{i_\mathrm{res}}] + \mathbb {E}[G_i]\right) + \frac{\tilde{G}_i (\nu _i)}{1-\tilde{G}_i (\nu _i)} \mathbb {E}[C]. \end{aligned}$$

The probability that a type-i customer arrives during a glue period of station i is $\mathbb {E}[G_i]/\mathbb {E}[C]$, and the probability that it arrives during a non-glue period equals $1-(\mathbb {E}[G_i]/ \mathbb {E}[C])$. Therefore,

$$\begin{aligned} \mathbb {E}[W_i]&= \frac{\mathbb {E}[G_i]}{\mathbb {E}[C]} \mathbb {E}[\bar{W_i}] + \frac{\mathbb {E}[C] - \mathbb {E}[G_i]}{\mathbb {E}[C]}\mathbb {E}[\tilde{W_i}] \\&\approx (1+\rho _i) \left( \frac{\mathbb {E}[G_i]}{\mathbb {E}[C]} \mathbb {E}[G_{i_\mathrm{res}}] + \frac{\mathbb {E}[C] - \mathbb {E}[G_i]}{\mathbb {E}[C]} \left( \mathbb {E}[C_{i_\mathrm{res}}] + \mathbb {E}[G_i]\right) \right) \\&\quad +\frac{\tilde{G}_i (\nu _i)}{1-\tilde{G}_i (\nu _i)} \left( \mathbb {E}[C] - \mathbb {E}[G_i] \right) . \end{aligned}$$

Let $R_{c_i}$ be the residual cycle time of the system with respect to station i. Then,

$$\begin{aligned} \mathbb {E}[R_{c_i}] = \frac{\mathbb {E}[G_i]}{\mathbb {E}[C]} \mathbb {E}[G_{i_\mathrm{res}}] + \frac{\mathbb {E}[C] - \mathbb {E}[G_i]}{\mathbb {E}[C]} \left( \mathbb {E}[C_{i_\mathrm{res}}] + \mathbb {E}[G_i]\right) , \quad i=1,\cdots ,N. \end{aligned}$$

We assume that $\mathbb {E}[R_{c_i}] =\mathbb {E}[R_{c}]$ for all $i=1,\ldots ,N$. We thus obtain (4.11):

$$\begin{aligned} \mathbb {E}[W_i] \approx (1+\rho _i) \mathbb {E}[R_c] + \frac{\tilde{G}_i (\nu _i)}{1-\tilde{G}_i (\nu _i)} \left( \mathbb {E}[C] - \mathbb {E}[G_i]\right) . \end{aligned}$$