On fairness in polling systems

Abstract

Which service discipline is more fair, exhaustive or gated? gated or globally-gated? How can we compare their fairness level? These questions are usually answered by handwaving as no fundamental research in this topic was yet conducted. Polling systems are widely used in many computer networks where several users compete for access to a common resource. Fairness is a fundamental aspect of polling systems; perhaps it serves as the motivation behind various polling schemes. Despite this fundamental role, fairness to customers in polling systems has not been extensively studied to date. This work is an attempt to model fairness in polling systems and study the relative fairness of various polling schemes. We focus, in the context of this work, on evaluating the system’s obedience to the FCFS policy as a measure of the fairness experienced by the individual customers. Using this metric we study the cyclic polling system under five different service disciplines: exhaustive, gated, binomial-gated, two-stage gated and globally-gated. For these systems we derive their “fairness” level both in a discrete system model and in a continuous model. We use the analysis as well as numerical examples to provide basic observations on fairness of polling systems.

Appendices

Appendix 1: Calculation of $E[X_i^*| \text{ service } \text{ period } j ]$ and $E[X_i^*| \text{ switch-over } \hbox {period} j]$ for gated discipline

The first and the second moments of the number of customers residing at the queues when queue i is polled, defined by Eqs. (2) and (3), were derived in Takagi (1986).

First, we derive $E[X_i^*| \text{ service } \text{ period } j ]$. To obtain $E[A_{i/j}^*]$ we observe that $A_{i/j}^*$ is the number of Poisson arrivals in a period whose duration is distributed as the residual life of the service period $S_j$. Thus, similarly to Eq. (15), $E[A_{i/j}^*]$ is given by:

$$\begin{aligned} E[A_{i/j}^*] = \frac{\lambda _i}{2} \frac{E[{S_j}^2]}{E[S_j]}. \end{aligned}$$

(80)

The expected service period is derived from the number of customers waiting at the polling instant and the average service time for a single customer:

$$\begin{aligned} E[S_j] = f_j(j)b_j. \end{aligned}$$

(81)

To derive the second moment of $S_j$ we use $V[S_j]$ from Takagi (1986):

$$\begin{aligned} E[S_j^2] = (E[S_j])^2 + V[S_j] = f_j(j,j) b_j^2 + f_j(j) b_j^{(2)}. \end{aligned}$$

(82)

Therefore,

$$\begin{aligned} E[A_{i/j}^*] = \frac{\lambda _i \Big (f_j(j,j) b_j^2 + f_j(j) b_j^{(2)}\Big )}{2f_j(j)b_j} . \end{aligned}$$

(83)

The derivation of $E[L_{i/j}^*]$ is similar to that of the Exhaustive service discipline, and we obtain the same formulas as in Eqs. (21) and (24):

$$\begin{aligned} E[L_{i/j}^*] = \left\{ \begin{array}{ll} \frac{f_j(i,j)}{f_j(j)}, &{}i \ne j \\ \frac{f_j(j,j)}{2f_j(j)} + 1, &{}i=j \end{array} \right. . \end{aligned}$$

(84)

By substituting Eqs. (83) and (84) in Eq. (11) we get:

$$\begin{aligned} E[X_i^*| \text{ service } \text{ period } j ] = \left\{ \begin{array}{ll} \frac{\lambda _i \big (b_j^2 f_j(j,j) + b_j^{(2)} f_j(j)\big )}{2f_j(j)b_j} + \frac{f_j(i,j)}{f_j(j)}, &{}i \ne j \\ \frac{\lambda _i \big (b_j^2 f_j(j,j) + b_j^{(2)} f_j(j)\big )}{2f_j(j)b_j} + \frac{f_j(j,j)}{2f_j(j)} + 1, &{}i=j \end{array} \right. . \end{aligned}$$

(85)

We derive $E[X_i^*| \text{ switch-over } \text{ period } j]$ similarly. For $i=j$ we observe that it consists of the sum of the number of Poisson arrivals in a period whose duration is distributed as the service period $S_j$, and the number of Poisson arrivals in a period whose duration is distributed as the residual life of the switch-over period $R_j$. Thus, we get:

$$\begin{aligned} E[X_i^*| \text{ switch-over } \text{ period } j] = \lambda _i E[S_j] + \frac{\lambda _i}{2} \frac{E[{R_j}^2]}{E[R_j]}, \quad (i = j). \end{aligned}$$

(86)

For $i \ne j$, $E[X_i^*| \text{ switch-over } \text{ period } j]$ additionally includes the customers that were waiting at queue j at the instant it was polled. Thus, we get:

$$\begin{aligned} E[X_i^*| \text{ switch-over } \text{ period } j] = f_j(i) + \lambda _i E[S_j] + \frac{\lambda _i}{2} \frac{E[{R_j}^2]}{E[R_j]}, \quad (i \ne j). \end{aligned}$$

(87)

Thus, from Eqs. (86), (87) and (81) we get:

$$\begin{aligned} E[X_i^*| \text{ switch-over } \text{ period } j] = \left\{ \begin{array}{ll} f_j(i) + f_j(j) b_j \lambda _i + \frac{\lambda _i r_j^{(2)}}{2 r_j}, &{}i \ne j \\ f_i(i) b_i \lambda _i + \frac{\lambda _i r_i^{(2)}}{2r_i}, &{}i=j \end{array} \right. \cdot \end{aligned}$$

(88)

Appendix 2: Notations and basic relations

$\tilde{C}$ :

An arbitrary customer arriving at the system.

$\tilde{C}_i$ :

An arbitrary customer arriving at queue i.

N :

Number of queues in the system.

$\lambda _i$ :

Intensity of the Poisson arrival to queue i.

$\lambda$ :

System arrival rate; $\lambda =\sum _{i=1}^{N}{\lambda _i}$.

$B_i$ :

Service time of a customer at queue i.

$b_i$, $b_i^{(2)}$ :

Mean and second moment of $B_i$.

$S_i$ :

Duration of the service period in queue i.

$R_i$ :

Duration of switch-over period from queue i.

$r_i$, $r_i^{(2)}$ :

Mean and second moment of $R_i$.

r :

$=\sum _{i=1}^{N}{r_i}$.

$\rho _i$ :

Utilization of queue i; $\rho _i=\lambda _i b_i$.

$\rho$ :

System utilization; $\rho = \sum _{i=1}^{N}{\rho _i}$.

$X_i^j$ :

Number of customers residing at queue j when queue i is polled.

$W_i$ :

Waiting time of a customer at queue i.

C :

Length of the server cycle.

$X_i$ :

Number of customers residing at queue i when it is polled; $X_i = X_i^i$.

$f_i(j)$ :

$E[X_i^j]$.

$f_i(j,k)$ :

$\left\{ \begin{array}{ll} E[X_i^j X_i^k], &{}j \ne k\\ E[X_i^j (X_i^j-1)], &{}j=k\\ \end{array} \right.$

$X_i^*$ :

Number of customers residing at queue i at an arbitrary moment.

$E[X_i^*| \text{ service } \text{ period } j ]$ :

Mean number of customers residing at queue i at an arbitrary epoch of the service period in queue j.

$X^*$ :

Number of customers residing at the entire system at an arbitrary moment.

$Z_{j,k}$ :

The event that an arbitrary observation point of a type-j service period corresponds to a service period in which $X_j=k$.

$Y_i^*$ :

Number of customers residing at the entire system, found by an arbitrary type-i customer arriving at an arbitrary moment, and served ahead of it (fair).

$E[Y_i^*| \text{ service } \text{ period } j ]$ :

Mean number of customers residing at the entire system, found by an arbitrary type-i customer arriving at an arbitrary epoch of the service period in queue j, and served ahead of it (fair).

$Y^*$ :

Number of customers residing at the entire system (out of $X^*$), found by an arbitrary customer arriving at an arbitrary moment, and served ahead of it (fair).

$U_i^*$ :

Number of customers residing at the entire system, found by an arbitrary type-i customer arriving at an arbitrary moment, and served after it (unfair).

$U^*$ :

Number of customers residing at the entire system (out of $X^*$), found by an arbitrary customer arriving at an arbitrary moment, and served after of it (unfair).

F :

The fairness level of the system, fraction of “the expected number of customers that were served ahead of a tagged arbitrary customer $\tilde{C}$” out of “the expected number of customers that were present at the system upon customer $\tilde{C}$’s arrival”; $F = \frac{E[Y^*]}{E[X^*]} = 1 - \frac{E[U^*]}{E[X^*]}$.

$F_{cont}$ :

The fairness level of the continuous polling system; $F_{cont} = 1-\frac{E[N_{slips}]}{E[N_{arrivals}]}$.

$L_{i/j}^*$ :

Number of customers residing at queue i at an arbitrary epoch of the service period in queue j, out of the ones who resided at the queue at the beginning of that service period.

$A_{i/j}^*$ :

Number of customers residing at queue i at an arbitrary epoch of the service period in queue j, out of the ones who arrived at the queue at during that service period.